Algebra I

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ALGEBRA I

Units 4 - 7

by

The Algebra I Development Team

iii

ALGEBRA I

UNIT 4 POWERS AND POLYNOMIALS . . . . . . . . . 1

4.0 Review . . . . . . . . . . . . . . . . 2 4.1 Properties of Exponents . . . . . . . . . . 4 4.2 Polynomials: Classifying and Simplifying. . . . . . 8 4.3 Adding and Subtracting Polynomials . . . . . . 10 4.4 Multiplying By a Monomial . . . . . . . . . 12 4.5 Factoring: Integers and Monomials . . . . . . 14 4.6 Common Monomial Factors . . . . . . . . . 16

UNIT 5 MORE FACTORING. . . . . . . . . . . . . 19

5.0 Review . . . . . . . . . . . . . . . . 20 5.1 Polynomial Products . . . . . . . . . . . 22 5.2 Factoring: x 2 + bx + c . . . . . . . . . 26 5.3 Factoring: ax 2 + bx + c and ax 2 + bxy + cy 2 . 28 5.4 Factoring Special Types . . . . . . . . . . 32 5.5 Factoring Combined Types . . . . . . . . . 36 5.6 Factoring By Grouping . . . . . . . . . . . 38 5.7 Solving Quadratic Equations by Factoring. . . . . 42 5.8 Consecutive Integer Problems . . . . . . . . . 46

UNIT 6 MULTIPLYING AND DIVIDING RATIONAL EXPRESSIONS 51

6.0 Review . . . . . . . . . . . . . . . . 52 6.1 Rational Expressions . . . . . . . . . . . 54 6.2 Rational Expressions: Simplest Form. . . . . . . 58 6.3 Using -1 in Factoring . . . . . . . . . . . 62 6.4 Dividing Powers . . . . . . . . . . . . . 66 6.5 Simplifying a Product . . . . . . . . . . . 70 6.6 Multiplying and Dividing . . . . . . . . . . 74 6.7 Dividing by a Monomial . . . . . . . . . . . 78 6.8 Dividing by a Binomial . . . . . . . . . . . 82

UNIT 7 ADDING AND SUBTRACTING RATIONAL EXPRESSIONS 87

7.0 Review . . . . . . . . . . . . . . . . 88 7.1 Adding Rational Expressions with Monomial

Denominators. . . . . . . . . . . . . . 90 7.2 Adding Rational Expressions with Polynomial

Denominators. . . . . . . . . . . . . . 96 7.3 Adding Other Types of Rationals . . . . . . . 100 7.4 Subtracting Rational Expressions . . . . . . . 104 7.5 Simplifying by Factoring Out -1 . . . . . . . . 108 7.6 Complex Rational Expressions . . . . . . . . 112 7.7 Complex Rational Expressions:

Polynomial Denominators . . . . . . . . . . 118

UNIT 4

POWERS AND POLYNOMIALS

Page

Lesson 4.0 Review . . . . . . . . . . . . . . . 2

Lesson 4.1 Properties of Exponents . . . . . . . . . 4

Lesson 4.2 Polynomials: Classifying and Simplifying. . . . . 8

Lesson 4.3 Adding and Subtracting Polynomials . . . . . 10

Lesson 4.4 Multiplying By a Monomial . . . . . . . . 12

Lesson 4.5 Factoring: Integers and Monomials . . . . . 14

Lesson 4.6 Common Monomial Factors . . . . . . . . 16

2

Lesson 4.0 Review Unit 4

Terminology: An open sentence contains one or more variables and is neither true

nor false until a replacement is made for the variable(s).

A solution is any replacement which makes an open sentence true.

Equivalent equations are those that have the same solution.

Properties for Equations: Addition: If a = b, then a + c = b + c. (also a - c = b - c)

Multiplication: If a = b and c ≠ 0, then ac = bc (also a = b)

c c

Solving Equations:

1. Simplify each side.

2. Use the addition property to get a variable term on one side and

a constant on the other.

3. Use the multiplication property to get the variable alone.

Example 1:

x - 4 = 12

x - 4 + 4 = 12 + 4 . Addition property.

x = 16

Example 2: 2x = 12

3

3 . 2x = 12 . 3 . Multiplication property.

2 3 2

x = 18

Example 3:

2(3x + 4) + x = 3x - 7 - x

6x + 8 + x = 2x - 7 . Simplify.

7x + 8 = 2x - 7 . Simplify.

7x + 8 - 2x = 2x - 7 - 2x . Addition property.

5x + 8 = -7 . Simplify.

5x + 8 - 8 = -7 - 8 . Addition property.

5x = -15 . Simplify.

5x = -15 . Multiplication property.

5 5 (Multiply by 1/5 or divide by 5.)

x = -3 . Answer.

3

Lesson 4.0

Words to Symbols:

Understanding key words and punctuation marks allows word

problems to be translated into equivalent algebraic equations.

(See Lesson 3.6.)

Solving Word Problems:

1. Choose a variable to represent what is to be found.

2. Plan and write the equation, using the facts in the problem.

3. Solve the equation and state the solution to the problem.

4. Check the solution with the facts of the problem.

Review Problems Solve each equation. Check the solution:

1. 3 = x - 12 9. 6a + 24 = 10a - 16

2. 2 + x = -15 10. x + 3 = 2x - 12

3. 3x = -54 11. 9 + 8t = 6t + 6

4. -x = -2(-5) 12. 3m + 7 + 7m = -3 + 5m

5. - 2 x = -16 13. 7 - x = 5x - 5 - 5x

3

14. -4(a - 3) = 2(a - 6)

6. -12 + 6x - 2x = 8

15. -4 - (8 + 2x) = -3(1 + 4x)

7. 10p - 42 - 2p = 38

16. 8r - 3(4 - 2r) = 6(r + 1) - 2

8. 2 = 3m + 18 - 5m

Write an equation for each problem; then solve it:

17. Two less than five times a number is 23. Find the number.

18. Joe's age is increased by 5 times his age. The result is 12. What is

Joe's age?

19. Separate 52 marbles into two groups so the second group has 2 less

than 5 times the number in the first group.

20. What are the dimensions of a rectangle which is twice as long as it is

wide if its perimeter is 48cm?

4

Lesson 4.1 Properties of Exponents Unit 4

Rules: In x4, x is the base and 4 is the exponent. This can be read

"the fourth power of x" or "x to the fourth power" or "x to the

fourth."

Product of Powers Rule: When multiplying powers with the

same base, add the exponents and keep the base.

am . an = am+n

Examples: 1. Simplify:

y3 . y4 = . Same base is "y".

y 3+4 = . Keep the base and add exponents.

y7 . (Think: y . y . y . y . y . y . y = y7.)

2. Simplify:

x6 . x = . Same base. Remember x = x1.

x6+1 = . Keep same base; add exponents.

x7 . Add 6 + 1.

3. Simplify:

x3 . y3 . Does not have the same base so it

cannot be simplified any further.

Practice: Simplify:

A. y2 . y4 =

B. y8 . y3 =

C. x . x7 =

D. x . y5 =

Rule: The Commutative and Associative Properties can be used to

regroup a problem so the product of powers rule can be

applied.

5

Lesson 4.1

Example: 4. Simplify:

(3y2)(2y4) =

(3 . 2)(y2 . y4) = . Group like factors.

6 . y2+4 = . Multiply like factors.

6y6 . Product of Powers.

Practice: Simplify:

E. (4x2)(2x5)

F. (3x2y4)(4xy3)

Example: 5. Simplify; then evaluate for x = -1:

(-x3)(3x2)(-2x4) =

(-1x3)(3x2)(-2x4) = . -x3 = -1x3.

(-1 . 3 . -2)(x3 . x 2 . x4) = . Regroup.

(6) (x3+2+4) = . Product of Powers.

6x9 = . Simplified answer.

6(-1)9 = . Substitute -1 for x.

6(-1) = . (-1)9 = -1.

-6 . Evaluated answer.

Rule: Power of a Power Rule: To simplify a power of a power,

multiply the exponents.

(am)n = am n.

Example: 6. Simplify:

(y3)4 =

y3 4 = . Multiply the exponents.

y12 . (Think: y3 . y3 . y3 . y3 = y12.)

Practice: Simplify:

G. (x4)6 =

H. (c5)7 =

Rule: Power of a Product Rule: To find the power of a product, apply

the exponent outside the grouping symbols to each factor

inside the grouping symbols.

(ab)m = am . bm

6

Lesson 4.1

Examples: 7. Simplify: 8. Simplify:

(2x)3 = (xy4)3 =

23 . x3 = x3 . (y4 ) 3 =

8x3 x3y12

Note: From the examples above, notice that each

exponent inside the grouping symbols is multiplied by the exponent outside the grouping symbols.

9. Simplify:

(-4x2)4 =

(-4)1 • 4 x2 • 4 = . Remember -4 = (-4)1.

(-4)4 . x8 = . Multiply exponents.

256x8 . Simplify: (-4)4 = 256.

Practice: Simplify:

I. (-3x4)2 =

J. (2y3)3 =

Example: 10. Simplify:

(3a3b2c)4 =

(31 . a3 . b2 . c1)4 = . 3 = 31 and c = c1.

(31)4(a3)4(b2)4(c1)4 =

34a12b8c4 = . Multiply each exponent by 4.

81a12b8c4 . Simplify: 34 = 81.

Practice: Simplify:

K. (4x2y3z)2 =

L. (-22d2e3f7)2 =

Example: 11. Simplify:

(4a2b)2(ab3)3(2a2b) =

(42a4b2)(a3b9)(2a2b) =

(16a4b2)(a3b9)(2a2b) =

(16 . 2)(a4 . a3 . a2)(b2 . b9 . b) = . b = b1.

32a9b12

7

Lesson 4.1

HOMEWORK

Simplify, if possible:

1. x2 . x4 8. (6z2)(-7z5)

2. c3 . c5 9. (-5a2)(-6a)

3. r4 . r 10. (6ab2)(2a3b5)

4. d3 . d8 11. (-bm5)(3b3m4)

5. (5x2)(3x3) 12. (5x2yz3)(-4y3)

6. (4y3)(2y) 13. (-3x)(-6y2)

7. (-7b2)(3b5) 14. (6x)(yz)

Simplify; then evaluate for x = -3, y = -4:

15. (-4x2)(3x)

16. (-3x2)(2y)

17. (2xy2)(3y3)

18. (6x2)(-3x3)(x)

Simplify:

19. (6y2)3 25. (-6a2b)2(ab2)3

20. (-3x2y3)4 26. (2xy)2(-x3y4)2

21. (-3a3b2c8)2 27. (x)2(xy)2

22. (x2y3z9)7 28. (x2y)2(-3x3y4)2

23. (5a2)2(a3)8 29. (de)2(2de)

24. (-2x3)2(4x2)3 30. (a2y)x(ay5)x

8

Lesson 4.2 Polynomials: Classifying and Simplifying Unit 4

Rule: A monomial is a term that is either a constant, a variable, or

a product of a constant and one or more variables.

Examples: Types of monomials:

1. 17 is a constant

2. x is a variable

3. 3 x4 is a product of a constant and a variable.

2

4. -6a2bc3 is a product of a constant and several variables.

Rule: The degree of a monomial is determined by finding the sum of

the exponents of the variable in the monomial. A non-zero

constant has a degree of 0 and the constant 0 has no degree.

Examples: Find the degree of each monomial:

5. 11y3 Degree: 3

6. 4x2y3 Degree: 5

7. 2a2n3r Degree: 6 . (6 = 2 + 3 + 1)

8. 7 Degree: 0

Practice: Find the degree of each monomial:

A. 11xy2 Degree:

B. 19 Degree:

C. 6a2x3y Degree:

D. -7rx5y Degree:

Rules: A polynomial is a monomial or the sum or difference of two or

more monomials.

Polynomials with one, two, or three terms have special

names:

Monomial: A polynomial with one term: 4y3

Binomial: A polynomial with two terms: 2x2 + 7y

Trinomial: A polynomial with three terms: 7x2 + 6x + 5

Practice: Label each polynomial as either a monomial, binomial, or

trinomial:

E. 3x2 + 2x - 6:

F. xy2 + 2x:

G. 8a2bc3:

9

Lesson 4.2

Rule: The degree of a polynomial in simple form is the same as the

highest degree of any of its terms.

When simplifying a polynomial, combine all like terms and

then write it in descending order of exponents (highest

exponent first, next highest exponent second, etc.).

Example: 9. Simplify:

6y2 - 8y3 + 2y2 + 4y - 8y3 + 4 =

-8y3 - 8y3 + 6y2 + 2y2 + 4y + 4 = . Regroup like terms.

-16y3 + 8y2 + 4y + 4 = . Combine like terms.

Degree: 3

HOMEWORK

What is the degree of each polynomial?

1. 4x7 4. x2y3z8

2. 8x 5. 2x2 - 3xy4

3. 11 6. x6 + 3x2y5 - 2y4

Classify each polynomial as a monomial, binomial, or trinomial:

7. 4x2 - 3y 10. 4b2 + 2x

8. 6a4 + 2x - 3y 11. 6x2 -3x + 4

9. 6d 12. 2(x2 + 3y3)

Simplify; write answers in descending order of exponents:

13. 8x2 - 3x + 4x - 2 20. 4ax2 - 6y + 5ax2 - 4y

14. 3y3 + 2y - 6y2 + 4y 21. 2b + 6ab2 - 3a2b + 7b

15. 3a + 4a2 - 6a + 2a2 22. 2a - 3a2 + 2a - 6ab

16. 7c + 6c2 + 5c3 - 3c2 23. 5a2b3 - 2ab + 6ab - 4a2b3

17. 6r + 2r2 - 6r - 2r2 + 9 24. 3xy2 + 2x2y - 6x2y2 + 2x2y - 4x2y2

18. ey2 + 2y3 + 6y - 4y2 + 2y 25. 6x2y + 2xy - 3x2y - xy - 2x2y - x2y

19. 6a + 4a - 3a2 + 2a - 5a2

10

Lesson 4.3 Adding and Subtracting Polynomials Unit 4

Rule: To add polynomials, group like terms in descending order of

exponents; then combine like terms.

Example: 1. Add:

(2x2 - 3x + 2) + (6x2 - 4x + 7) =

2x2 + 6x2 - 3x - 4x + 2 + 7 = . Group like terms,

descending order).

8x2 - 7x + 9 . Combine like terms.

Practice: Add:

A. (3y3 + 4y - 2) + (5y3 + 2y - 8)

B. (5a2 + 3a3 - 4a + 2) + (3a5 + 6a - 5)

Rule: To subtract polynomials, add the opposite of each term of

the polynomial that is to be subtracted. (This is the same as

distributing -1.)

Example: 2. Subtract:

(a2 + 3a - 4) - (6a2 - 4a + 5) =

a2 + 3a - 4 - 6a2 +4a - 5 = . Add the opposite.

a2 - 6a2 + 3a + 4a - 4 - 5 = . Group like terms.

-5a2 + 7a - 9 . Combine like terms.

Practice: Subtract:

C. (3y3 - 2y2 + 4y - 5) - (3y2 + 6y + 4)

D. (2a4 + 3a3 - 5a + 2a2) - (7a2 + 4a3 - a + 7)

11

Lesson 4.3 HOMEWORK

Add:

1. (7x2 - 3x + 2) + (-8x2 + 2x + 6)

2. (b4 - 7b + 9) + (b5 - b4 + 2b3)

3. (2y5 - y4 + 3y3) + (y4 - 9y + 5)

4. (2c2 - 3c + 5) + (3c2 - 7c + 4)

Subtract:

5. (4x2 + 3x - 2) - (7x + 7)

6. (5y2 + 3y - 4) - (6y + 8)

7. (2x4 - 3x2 + 2x) - (3x3 + 4x2 - 5)

8. (a5 - 2a + 4a2 - 5) - (6a5 - 3a3 + 2a2 - 7)

9. (7d3 + 5d - 4) - (5d3 + 4d - 4)

10. (7x3 + 3x2 + 7 - 2x) - (-2x + 3x2 + 7 + 7x3)

Perform the indicated operation:

11. (2x2 - 3x + 4) + (5x + 3)

12. (2x + 4) + (3x2 - 5x3 + 4x - 2)

13. (6x3 - 4x - 3) - (5x3 + 4x2 - 3x + 5)

14. (2y + 3y2 - 5) + (y2 + 3y - 2)

15. (5a4 - 3a2 + 4a3 - 4) - (6a5 - 2a + 4)

16. (5c3 + 3) - (7c2 + 4c3 - 3c2 + c - 8)

17. (13c2 + 7c + 5) + (-7c - 3 - 7c2 - 2 - 6c2)

18. (8y2 + 2y - 5) - (6y3 + 3y - 2)

19. (9x + 4x2 - 3x3) - (6y3 + 3y - 2)

20. (21x - 3x2 + 7x) + (7x2 - 3x + 14)

12

Lesson 4.4 Multiplying by a Monomial Unit 4

Rule: Some polynomials can be simplified by first using the

Distributive Property and then combining like terms.

Example: 1. Simplify:

x2 + 6x + 8 + 4(x2 + 3x - 4) =

x2 + 6x + 8 + 4x2 + 12x - 16 = . Distribute the 4.

x2 + 4x2 + 6x + 12x + 8 - 16 = . Group like terms.

5x2 + 18x - 8 . Combine like terms.

Practice: Simplify:

A. y2 + 4y + 3(2y2 + 3y - 2)

B. 3a2 + 4a + 5(a2 - 2a - 5)

Rule: To simplify the product of a monomial and a polynomial, use

the Distributive Property and the Product of Powers Property.

Example: 2. Multiply:

2a2(5a4 + 6a3 + 2a) =

(2a2 . 5a4) + (2a2 . 6a3) + (2a2 . 2a) = . Distribute the 2a2.

10a6 + 12a5 + 4a3

Practice: Multiply:

C. x2(3x3 - 2x2 + 4) D. c3(4c5 + 6c3 - 3c + 5)

Rule: The same steps are followed when there are two or more

variables.

Example: 3. Multiply:

a2c3(3a2 + 4ac3 - 2a + 3c - 5)

(a2c3.3a2) + (a2c3.4ac3) - (a2c3.2a) + (a2c3.3c) - (a2c3.5)

. Distribute the a2c3.

3a4c3 + 4a3c6 - 2a3c3 + 3a2c4 - 5a2c3

. Group like terms.

13

Lesson 4.4

4. Multiply:

2xy(3x2 - 4xy2) - 3x(2x3y + 4x2 - 7) =

(2xy.3x2) - (2xy.4xy2) + (-3x.2x3y) + (-3x.4x2) - (-3x. 7) =

6x3y - 8x2y3 - 6x4y - 12x3 + 21x =

-6x4y + 6x3y - 12x3 - 8x2y3 + 21x

Practice: Multiply:

E. xy(3x4 + 4y - 2xy2) F. 2s2t3(4st + 3s2 - 2t3 + 5)

HOMEWORK

Simplify: Multiply:

1. x2 + 3x - 5(x2 + 5x + 3) 5. r2(3r4 - 2r2 + 4)

2. y3 + 3y2 + 2(y3 - 3y2 + 7y - 4) 6. 2x(5x4 - 3x2 + 4)

3. p + p2 - 5(p2 + 3p - 4) 7. a3(a4 - 3a2 + 6a - 3)

4. c4 - 2c + 4(c3 + c2 - 4c + 2) 8. 3x3(3x3 + 2x2 + x - 5)

Simplify:

9. x2y3(3x2 + 4y2 - 3xy3)

10. ab2(3ab + 2a2 - 3ab3 - 4)

11. b3d2(3bd2 + 2b - 3d - 4d3)

12. -xy3(-3x3 + y2 - 4xy + 2)

13. 3a3c4(2a4 - 3c2 + 2ac + 6)

14. -3xy5(2xy4 - 3xy + 4y3 + 2)

15. 3qr(4r2 + 3r - 2qr3) - 6qr(2r2 + 3qr3 - 5r)

16. -6a3b2(3a4b - 2a2b + 2) - 4a3b2(-2a4b + 3a2b - 8)

17. xy2(3xy4 + 2x2y - 5y) - xy2(2x2y - 3y - 6xy4)

18. m2r3(3mr + 2mr2 - 6m2r) + m2r3(-3mr - 2mr2 + 6m2r)

15

Lesson 4.5

Note: Use a similar approach to find a missing factor with a

numerical coefficient.

Example: 3. Find the missing factor:

(6x3)(?) = -24x8

(6x3)(?) = -24 . x8

6 . (?) . x3 . x? = -24 . x8 • 6 • (-4) = -24

6 . (-4) . x3 . x5 = -24 . x8 • x3 • x5 = x8

6x3 (-4x5) = -24x8

Practice: Find the missing factor:

E. (3y2)(?) = 9y7 F. (?) (5m3) = 35m8

HOMEWORK

Factor each number into primes, if possible:

1. 14 4. 48

2. 21 5. 38

3. 17 6. 29

Find the missing factor:

7. (y3)(?) = y4 14. (11r4)(?) = 44r8

8. (?)(a7) = a13 15. (?)(16y) = 16y2

9. (?)(c) = c5 16. (3ab)(?) = 3a2b

10. (r4)(?) = r11 17. (6xy)(?) = 24x2y4

11. (4x2)(?) = 8x3 18. (?)(5r2s3) = 15r7s11

12. (?)(6a3) = 24a5 19. (?)(4x2y5) = 28x5y7

13. (?)(5c4) = 20c9 20. (x8y2z5)(?) = 21x9y3z5

14

Lesson 4.5 Factoring: Integers and Monomials Unit 4

Rules: A prime number is a whole number greater than one whose

only factors are 1 and the number itself.

The prime factorization of a number is the result of rewriting

the number as the product of its prime factors. There is only

one prime factorization of any given number.

Example: 1. Factor 36 into primes:

36 or 36

4 . 9 12 . 3

2 . 2 . 3 . 3 4 . 3 . 3

22 . 32 2 . 2 . 3 . 3

22 . 32

Note: Notice that there is only one prime factorization

of 36, 22 . 32, but it can be found in several different ways.

Rule: By using the Product of Powers Property, a missing factor can

be found.

Example: 2. Find the missing factor:

x4 . ? = x11

x4 . x? = x11

x4+? = x11

x4+7 = x11

Thus, the missing factor is x7.

Practice: Factor each number into primes:

A. 24 B. 32

Find the missing factor:

C. y4 . ? = y7 D. ? . a9 = a13

16

Lesson 4.6 Common Monomial Factors Unit 4

Rule: To factor a polynomial into the product of a monomial and a

polynomial, the greatest common factor (GCF) must be

found.

Example: Factor out the greatest common monomial:

1. 3x2 + 6x + 12 =

3(x2) + (3)(2)(x) + (3)(2)(2) = . 3 is the greatest common

3(x2 + 2x + 4) factor.

2. 3a2 + 12a - 3 =

3(a2) + (2)(2)(3)(a) + 3(-1) = . -3 = 3 . -1.

3(a2 + 4a - 1) . 3 is the greatest common

factor.

Practice: Factor out the greatest common monomial:

A. 4x2 + 16x + 8 B. 5y2 - 15y - 5

Example: 3. Factor out the GCF:

15x2 - 45x + 30 =

3 . 5 . x2 - 3 . 3 . 5 . x + 2 . 3 . 5 = . Factor 15, 45, and 30

15(x2 - 3x + 2) into primes.

. The GCF is 3 . 5 or 15.

4. Factor out the GCF:

x3 - 5x2 + x =

(x)(x)(x) - 5(x)(x) + (x) = . The GCF is x.

x(x2 - 5x + 1) . Remember that 1 . x = x.

Practice: Factor out the GCF:

C. 10x2 + 30x - 40 D. y3 + 3y2 - y

Example: 5. Factor out the GCF:

3y4 + 12y3 - 15y2 =

3(y4 + 4y3 - 5y2) = . Find the greatest common whole

number factor, 3.

3y2(y2 + 4y - 5) . Now look for the greatest common

variable factor, y2. The GCF is 3y2.

17

Lesson 4.6

Practice: Factor the GCF from each:

E. 4x5 - 12x3 + 6x2 F. 12y6 - 6y4 - 18y

Note: Remember these steps when factoring out the

greatest common monomial factor: 1. Factor out the Greatest Common Numerical Factor, if

possible 2. Factor out the Greatest Common Variable Factor, if

possible.

HOMEWORK

Factor out the Greatest Common Factor:

1. 3y2 + 6y - 9 16. 4x3 - 16x2

2. 4x2 - 6x + 12 17. 6r9 - 3r3 + 3r

3. 5y2 - 10y + 20 18. 15x9 + 45x6 - 75x3

4. 2a2 - 12a + 16 19. 7a5 - 35a

5. 4c2 - 18c - 22 20. 4a5 + 3a4 - 2a3 + a

6. 7x2 - 21x + 14 21. 16a5 - 20a2 + 4

7. 9y2 - 27y - 18 22. 4x3 - 20x2 + 24x

8. 8a2 + 16a - 8 23. 4t3 - 34t

9. 21d2 + 42d + 21 24. 4a4 - 3a3 + a2

10. 17y2 + 34y + 51 25. 21y2 + 7y - 14

11. 18x3 - 7x2 + 14x 26. 78x2 + 104x - 52

12. 3c5 + 21c3 - 4c2 27. 12x2 - 16x + 24

13. 8r8 + 6r6 - 4r4 28. 64x5 - 112x4 + 98

14. 32b4 - 3b2 + b 29. 425a2 - 123a + 25

15. 21m7 - 7m4 + 21m2 30. 62r7 - 155r5 - 186

 

UNIT 5

MORE FACTORING

5.0 Review. . . . . . . . . . . . . . . . . . . 20

5.1 Polynomial Products . . . . . . . . . . . . . . 22

5.2 Factoring: x 2 + bx + c . . . . . . . . . . . . 26

5.3 Factoring: ax 2 + bx + c and ax 2 + bxy + cy 2 . . . . 28

5.4 Factoring Special Types . . . . . . . . . . . . . 32

5.5 Factoring Combined Types . . . . . . . . . . . . 36

5.6 Factoring By Grouping . . . . . . . . . . . . . 38

5.7 Solving Quadratic Equations by Factoring. . . . . . . . 42

5.8 Consecutive Integer Problems . . . . . . . . . . . 46

20

Lesson 5.0 Review Unit 5 Properties of Exponents:

1. Product of Powers : am . an = am+n

Example : x2 . x3 = x2+3 = x5

2. Power of a Power : (am)n = amn

Example : (x2)3 = x2 3 = x6

3. Power of a Product : (ab)m = ambm

Example : (3x)2 = (32)(x2) = 9x2

Terminology: Monomial: 4xy2 (1 term)

Binomial: 2x2 + 7y (2 terms)

Trinomial: 7x2 + 6x + 5 (3 terms)

Polynomial: A monomial or the sum or difference of two or more

monomials.

Degree: 3x2y3 + 4x3y + 2xy . This is a 5th degree

polynomial.

5th degree 4th degree 2nd degree

term term term

Adding and Subtracting: To add polynomials, group and combine like terms in descending

order of exponents.

To subtract a polynomial, add the opposite of each of its terms.

Example: (7a2 + 3a - 4) - (2a2 - 5a + 6) =

7a2 + 3a - 4 - 2a2 + 5a - 6 = . Add the opposite.

7a2 - 2a2 + 3a + 5a - 4 - 6 = . Group like terms.

5a2 + 8a - 10 . Combine like terms.

Multiplying by a Monomial: Use the Distributive Property and the properties of exponents.

2a2b(3a + 2a2b2 + 5b3) = 6a3b + 4a4b3 + 10a2b4

= 4a4b3 + 6a3b + 10a2b4

Common Monomial Factors: 1. Find the greatest common numerical factor.

2. Find the greatest common variable factor.

3. Use the Distributive Property to "undo" multiplication:

6a3 + 4a4b3 + 10a3b4 = 2a3 (3 + 2ab3 + 5b4)

21

Lesson 5.0

Review Problems

Simplify, if possible:

1. x5 . x4 4. (2x3)4

2. (2x3)(3x5) 5. (3x2y)(-5x3y4)2

3. (x4)5

Simplify each polynomial; then tell its degree:

6. 6r + 2r2 - 6r + 9

7. 2b + 6a2b3 - 3a3b2 + 7b

8. 7a2b3 - 2ab + 6ab - 4a2b3

Simplify:

9. (2x4 - 3x2 + 5x) + (5x3 - 3x2 + 2)

10. (6x3 - 4x - 3) - (5x3 + 4x2 - 3x + 5)

11. (8x - 3x3 + 4x4) - (6x3 + 3x - 2)

12. m4 - 2m + 4(m3 + m2 - 4m + 2)

13. x3y2(3xy2 + 2x - 3y - 4y3)

14. 2ab(4b2 + 3b - 2ab3) - 6ab(2b2 + 3ab3 - 5b)

Find the missing factor(s):

15. (4x2)(?) = 12x6

16. (?)(4a2b5) = 28a5b7

Factor completely:

17. 3x2 + 6x + 12

18. 4a5 - 12a3 + 6a2

19. 21x3 + 7x2 - 14x

20. 32r8 - 16r6 + 64r4

22

Lesson 5.1 Polynomial Products Unit 5

Rule: The Distributive Property makes it possible to multiply two

binomials in the same way a monomial is multiplied by a

binomial.

Examples: 1. Multiply: 2x(4x + 3) = . Monomial x Binomial.

2x . 4x + 2x . 3 = . Distribute 2x.

8x2 + 6x . Multiply.

2. Multiply: (2x + 1)(4x + 3) = . Binomial x Binomial.

If a = 2x + 1, this problem becomes:

a(4x + 3) = . Substitute "a" for 2x + 1.

a(4x) + a(3) = . Distribute "a".

4ax + 3a . Multiply.

In the same way,

(2x + 1)(4x + 3) =

(2x + 1)4x + (2x + 1)3 = . Distribute 2x + 1.

(2x)4x + (1)4x + (2x)3 + (1)3 = . Distribute 4x and 3.

8x2 + 4x + 6x + 3 = . Multiply.

8x2 + 10x + 3 . Combine like terms.

Rule: The FOIL method makes multiplication easier. It suggests

multiplying First terms, Outer terms, Inner terms, and Last

terms.

Examples: 3. Multiply: (2x + 1)(4x + 3)

2x . 4x = 8x2 . First terms.

2x . 3 = 6x . Outer terms.

1 . 4x = 4x . Inner terms.

1 . 3 = 3 . Last terms.

8x2 + 10x + 3 . Combine like terms.

23

Lesson 5.1

4. Multiply: (5m - 4p)(2m + 3p)

5m . 2m = 10m2 . First terms.

5m . 3p = 15mp . Outer terms.

-4p . 2m = -8mp . Inner terms.

-4p . 3p = -12p2 . Last terms.

10m2 + 7mp - 12p2 . Combine like terms.

Practice: Multiply:

A. (2x + 3)(3x + 4)

B. (4x - 1)(4x + 1)

Rule: The FOIL method may be used to square a binomial. First,

write the power in expanded form, then multiply using FOIL.

Example: 5. Multiply: (3x + 2)2

(3x + 2) (3x + 2) . Product of two binomials.

9x2 . F

6x . O

6x . I

4 . L

9x2 + 12x + 4 . Combine like terms.

Example: A quick method of squaring a binomial:

6. Multiply: (4x - 5y)2

(4x)2 . Square the first term.

16x2 + 2(4x) (-5y) . Double product of the two

terms.

16x2 + 2(-20xy) + (-5y)2 . Square the last term.

16x2 - 40xy + 25y2 . Simplify each term.

24

Lesson 5.1

Practice: Square each binomial:

C. (x + 4)2

D. (2x - 3)2

Rule: The Distributive Property allows multiplication of any two

polynomials. In order, multiply the first polynomial by each

term of the second polynomial.

Examples: 7. Multiply: (2x - 1)(3x2 - 2x + 5)

(2x - 1)(3x2) = 6x3 - 3x2 . Multiply polynomial by 3x2.

(2x - 1)(-2x) = -4x2 + 2x . Multiply polynomial by -2x.

(2x - 1)(5) = 10x - 5 . Multiply polynomial by 5.

(6x3-3x2)+(-4x2+2x)+(10x-5) . Add the resulting products.

6x3 - 7x2 + 12x - 5 . Combine like terms.

8. Multiply: (5x + 3)(x2 + 4x - 2)

(5x + 3)(x2) = 5x3 + 3x2 . Multiply polynomial by x2.

(5x + 3)(4x) = 20x2 + 12x . Multiply polynomial by 4x.

(5x + 3)(-2) = -10x - 6 . Multiply polynomial by -2.

(5x3+3x2)+(20x2+12x)+(-10x-6) . Add the resulting products.

5x3 + 23x2 + 2x - 6 . Combine like terms.

Rule: For a simplified method of multiplying two polynomials,

multiply each term of one polynomial by each term of the

other polynomial; then combine like terms.

Example: 9. Multiply: (5x + 3)(x2 + 4x - 2) . Same as Example 8.

5x(x2) + 5x(4x) + 5x(-2) + . Multiply each term by 5x.

3(x2) + 3(4x) + 3(-2) = . Multiply each term by 3.

5x3 + 20x2 + 3x2 - 10x + 12x - 6 = . Simplify.

5x3 + 23x2 - 2x - 6 . Combine like terms.

25

Lesson 5.1

HOMEWORK

Multiply:

1. (3x + 1)(x + 2) 11. (r - 2p)2

2. (x + 4)(2x - 3) 12. (3r - 4x)2

3. (2a + 1)(3a + 5) 13. (2a - 3b)(4a + 5b)

4. (r - 3)(r + 3) 14. (2x + y)(3x - 2y)

5. (4m + 5)(4m - 5) 15. (2p - 3)(p + 1)

6. (2a - 1)(3a - 2) 16. (2r - 3)(2r - 3)

7. (5m + 2)(5m + 2) 17. (2x - 3y)(2x + 3y)

8. (2p + 1)2 18. (3a - 1)(2a2 + 5a - 7)

9. (3r - 4)2 19. (2x + 3)(x2 - 4x - 5)

10. (m - 3)2 20. (3m + 2)2(2m - 1)

26

Lesson 5.2

Factoring x2 + bx + c Unit 5 Rule: Multiplying two binomials often results in a trinomial.

Factoring such a trinomial reverses the process to obtain the two

binomials. For example, since (x + 3)(x + 4) = x2 + 7x + 12, then the factored form of x2 + 7x + 12 is (x + 3)(x + 4).

Example: 1. Factor: x2 + 5x + 6

x2 + 5x + 6 = (x )(x ) . Factor the first term.

x2 + 5x + 6 = (x + ?)(x +?) . Factor the last term.

. Think of the integer pairs

whose product is 6: 1 . 6;

-1 . -6; 2 . 3; -2 . -3. 3x

x2 + 5x + 6 = (x + 2)(x + 3) . Select the integer pair

2x whose sum is 5: 2 + 3 = 5.

Thus, x2 + 5x + 6 = (x + 2)(x + 3).

2. Factor: x2 - 5x + 6

x2 - 5x + 6 = (x )(x ) . Factor the first term.

x2 - 5x + 6 = (x + ?)(x + ?) . Factor the last term. Think of the integer pairs

whose product is 6: 1 . 6; -1 . -6;

2 . 3; -2 . -3.

-3x

x2 - 5x + 6 = (x - 2)(x - 3) . Select the integer pair

-2x whose sum is -5: -2 + (-3) = -5.

Thus, x2 - 5x + 6 = (x - 2)(x - 3). Note: When the third term is positive, (+), the two factors

must have the same sign as the middle term.

Example: 3. Factor: x2 + 5x - 24

x2 + 5x - 24 = (x )(x ) . Factor the first term.

x2 + 5x - 24 = (x + ?)(x + ?) . Factor the last term.

Think of the integer pairs

whose product is -24: 24 . -1;

-24 . 1; 12 . -2; -12 . 2; 8 . -3; -8 . 3;

6 . -4; -6 . 4. -3x

x2 + 5x - 24 = (x + 8)(x - 3) . Select the integer pairs whose

8x sum is +5: 8 + (-3 )= +5.

Thus, x2 + 5x - 24 = (x + 8)(x - 3).

27

Lesson 5.2 4. Factor: x2 - 5x - 24

x2 - 5x - 24 = (x )(x ) . Factor the first term.

x2 - 5x - 24 = (x + ?)(x + ?) . Factor last terms.

Write the integer pairs whose

product is -24: 24 . -1; -24 . 1;

12 . -2; -12 . 2; 8 . -3; -8 . 3; 6 . -4;

3x -6 . 4.

x2 - 5x - 24 = (x - 8)(x + 3) . Select the integer pair whose

-8x sum is -5: -8 + 3 = -5.

Thus, x2 - 5x - 24 = (x - 8)(x + 3). Note: When the third term is negative, (-), the two factors

must have opposite signs, and the factor with the larger absolute value will have the same sign as the middle term.

Practice: A. Factor: m2 - 2m - 15

HOMEWORK

Factor each trinomial into two binomials:

1. x2 + 6x + 8 14. r2 - 5r + 36 2. a2 + 12a + 20 15. x2 + 10x + 24

3. m2 + 8m + 12 16. a2 - a - 20

4. g2 + 4g - 12 17. b2 - 5b + 6

5. x2 - 4x - 12 18. x2 - 2x - 48 6. r2 - 12r + 20 19. m2 + 6m - 55

7. n2 - 6n + 9 20. x2 - 12x - 64

8. m2 - m + 12 21. x2 - 9x + 20

9. a2 + 7a - 18 22. m2 - 11m + 28

10. x2 - 3x - 28 23. z2 - 12z - 64 11. c2 + 4c - 21 24. a2 - 6a - 27

12. m2 + 9m + 8 25. x2a + 10xa + 25

13. d2 - 15d + 36

28

Lesson 5.3

Factoring ax2 + bx + c and ax2 + bxy + cy2 Unit 5

Rule: The same trial and error process used to factor trinomials of

the form x2 + bx + c (with an x2 coefficient of 1) is also used to

factor trinomials in which the x2 coefficient is not 1:

ax2 + bx + c, where "a" is a positive integer.

Example: 1. Factor: 3x2 - 14x + 15

3x2 - 14x + 15 = (3x )(x ) . Factor the first term.

(3 is prime, so the only possible factors are 3x and x.)

3x2 - 14x + 15 = (3x ?)(3x ?) . Factor the last term.

. Write the integer pairs whose

product is 15: 15 . 1; -15 . -1;

-9x 5 . 3; -5 . -3.

3x2 - 14x + 15 = (3x - 5)(x - 3) . Select the integer pairs whose

- 5x Inner and Outer products have a sum of -14x: -5x + -9x = -14x.

Thus, 3x2 - 14x + 15 = (3x -5)(x - 3).

Practice: A. Factor: 5x2 - 13x + 6

Rule: If the x2 coefficient is not prime, different combinations of it

may also need to be tried.

Example: 2. Factor: 6k2 + 11k - 10 . Note negative third term.

6k2 + 11k - 10 = (6k )(k ) . Factor the first term.

Possible factors are 6k .k or

3k . 2k. Try 6k . k first.

6k2 + 11k - 10 = (6k ?)(k ?) . Factor the last term. Try

(6k + 10)(k - 1) factors of -10 whose Inner and

(6k - 1)(k + 10) Outer products have a sum of

(6k - 5)(k + 2) 11k. None of these works.

(6k - 2)(k + 5)

Since the above combinations do not give a middle term of

+11k and reversing the signs gives a negative middle term, try

combinations using 3k and 2k as factors of 6k2.

29

Lesson 5.3

6k2 + 11k - 10 = (3k ?)(2k ?)

(3k -1)(2k +10) . Again, try factors of -10 whose

(3k+10)(2k -1) Inner and Outer products have

(3k + 5)(3k - 2) a sum of +11k.

(3k - 2)(2k + 5) . This combination works.

So, the factorization of 6k2 + 11k - 10 is (3k - 2)(2k + 5).

Practice: B. Factor: 4x2 + 4x - 15

Rule: With practice, the correct combination of factors may be found

without trying every possibility. Think of all possible factors of

first and last terms; the signs of the middle and last terms

may eliminate half of the combinations.

Example: 3. Factor: 18x2 - 9x - 20

Some possible factors: Middle term:

(18x - 20)(x + 1) -2x

(18x + 10)(x - 2) -26x

(9x - 10)(2x + 2) -2x

(9x + 5)(2x - 4) -26x

(6x - 10)(3x + 2) -18x

(6x - 20)(3x + 1) -54x

(6x + 4)(3x - 5) -18x

(6x + 5)(3x - 4) -9x Correct term.

Although other possible factors could be tried, since -9x is the

correct middle term, 18x2 - 9x - 20 = (6x + 5)(3x - 4).

4. Factor: 16x2 + 14x - 15

Some possible factors: Middle term:

(16x - 15)(x + 1) x

(16x - 1)(x + 15) 239x

(16x - 5)(x + 3) 43x

(16x - 3)(x + 5) 77x

(8x + 15)(2x - 1) 22x

(8x - 1)(2x + 15) 118x

(8x - 5)(2x + 3) 14x Correct term.

So, 16x2 + 14x - 15 = (8x - 15)(2x - 3).

30

Lesson 5.3

Rule: Factoring a polynomial with two variables is done in the same

way a polynomial with one variable is factored: Factor out the

GCF; then look for combinations of factors of the first and last

terms that give the correct middle term.

Examples: 5. Factor completely: 40r3 - 14r2t - 12rt2

40r3 - 14r2t - 12rt2 . 2 and r are both common factors.

2r(20r2 - 7rt - 6t2) . 2r is the GCF.

2r(4r )(5r ) . Try 4r and 5r as factors.

2r(4r - 3t)(5r + 2t) . Try combinations of factors of -6t2:

-3t and + 2t are correct.

-15rt . -15rt + 8rt = -7rt, the correct middle term.

+ 8rt

So, the complete factorization of 40r3 - 14r2t - 12rt2 is

2r(4r - 3t)(5r + 2t).

Notice that to have a middle term with both variables, an "r" is

kept with each factor of 20r2 and a "t" is kept with each factor

of -6t2.

6. Factor completely: 12k2 - 75m2

12k2 - 75m2 . 3 is a common factor.

3(4k2 - 25m2) . 3 is the GCF.

3[(2k)2 - (5m)2] . 4k2 - 25m2 is a difference of two squares.

3(2k + 5m)(2k - 5m) . A difference of two squares factors into a

sum and a difference.

+10km . 10km - 10km = 0; Inner and Outer

products must add to 0.

- 10km

So, the complete factorization of 12k2 - 75m2 is

3(2k + 5m)(2k - 5m).

7. Factor completely: 405x5 - 80xy4

405x5 - 80xy4 . 5 and x are common factors.

5x(81x4 - 16y4) . 5x is the GCF.

5x[(9x2)2 - (4y2)2] . 81x4 and 16y4 are perfect squares.

5x(9x2 + 4y2)(9x2 - 4y2) . A difference of two squares factors

into a sum and a difference.

5x(9x2 + 4y2)(3x+ 2y)(3x- 2y) . 9x2 - 4y2 is a difference of two

squares, but 9x2 + 4y2 is not. So, the complete factorization of 405x5 - 80xy4 is

5x(9x2 + 4y2)(3x + y)(3x - y).

31

Lesson 5.3 HOMEWORK

Factor each trinomial into two binomials:

1. 3x2 + 11x + 10 6. 12m2 + m - 20

2. 5x2 - 17x + 6 7. 15y2 - 8y - 12

3. 4r2 + 5r - 6 8. 6n2 - 13n - 15

4. 4p2 + 9p + 5 9. 14m2 - 15m - 9

5. 6m2 - m - 1 10. 10 + 5a - 5a2

Factor completely, if possible:

11. 3x2 + 3xy - 18y2

12. m2 - 4p2

13. 4a2 - r2

14. r2 - 12rx + 35x2

15. 2m2 + 20mp + 18p2

16. 25a2 - 16r2

17. 49x2 - 81y2

18. 16p2 - 16q2

19. 6k2 + 13km - 5m2

20. 10a2 - 3ab - b2

21. 8g2 + 4gf - 30f2

22. 6a2 - 9ab - 15b2

23. 6cd3 - 24c3d

24. 10m3n + 21m2n2 - 9mn3

25. 6a3b + 7a2b2 - 20ab3

32

Lesson 5.4 Factoring Special Types Unit 5

Rules: Some polynomials have a special pattern. Recognition of this

pattern makes them easier to factor. This lesson shows two of

these special cases: perfect trinomial squares and the

difference of two squares.

Trinomials of the type a2 + 2ab + b2 or a2 - 2ab + b2 are called

perfect trinomial squares. Factoring a perfect trinomial square

gives the square of a binomial.

a2 + 2ab + b2 = (a + b)2

a2 - 2ab + b2 = (a - b)2

Examples: 1. Factor: x2 + 10x + 25

x2 + 2(5x) + 52 . Can it be written in the a2 + 2ab + b2

form?

x2 + 2(5x) + 52 . First and last terms are perfect

squares.

. Middle term is twice the product of x and 5.

(x + 5)2 . Square of a binomial sum.

So, x2 + 10x + 25 = (x + 5)2.

2. Factor: 9x2 + 12x + 4

(3x)2 + 2(3x)(2) + 22 . Write it in a2 + 2ab + b2 form.

(3x)2 + 2(3x)(2) + 22 . First and last terms are perfect

squares.

. Middle term is twice the product of 3x and 2.

(3x + 2)2 . Answer.

3. Factor: 25x2 - 30x + 9

(5x)2 - 2(5x)(3) + 32 . Rewrite. Note negative middle term.

(5x - 3)2 . Answer. Note negative sign.

4. Factor: 16x2 - 40xy + 25y2

(4x)2 - 2(4x)(5y) + (5y)2 . Rewrite. Note negative middle term.

(4x - 5y)2 . Answer. Square of a difference.

33

Lesson 5.4

Practice: A. x2 - 14x + 49 B. 25x2 + 90x + 81

Rule: Factoring the difference of two squares gives a product of two

binomials. The binomial is a sum and a difference of the

square roots of the two squares:

a2 - b2 = (a + b)(a - b)

Examples: 5. Multiply: (x + 2)(x - 2) . Use FOIL.

x2 -4 . First and last products give x2 - 4.

(x + 2) (x - 2)

2x . Inner and Outer products give

2x - 2x = 0x.

-2x

So, the product is a binomial, the difference of two squares:

(x + 2)(x - 2) = x2 - 4.

6. Factor: x2 - 4

(x )(x ) . Factor first term.

(x + )(x - ) . Positive . negative = negative.

(x + 2)(x - 2) . Square root of last term is 2.

7. Factor: x2 - 36y2

(x )(x ) . Factor first term.

(x + )(x - ) . Positive . negative = negative.

(x + 6y)(x - 6y) . Square root of last term is 6y.

Practice: C. Factor: 16m2 - 9r2

D. Factor: 1.69a2 - 49b2

34

Lesson 5.4

HOMEWORK

Factor the perfect trinomials:

1. x2 - 14x + 49

2. x2 + 16x + 64

3. y2 - 6y + 9

4. a2 - 12a + 36

5. x2 - 16x + 64

6. x2 + 14x + 49

7. x2 + 2x + 1

8. t2 + 18t + 81

9. 9t2 - 6t + 1

10. 4n2 + 4nt + t2

11. 25d2 - 10d + 1

12. x4 + 2x2 + 1

13. 25y4 - 10y2x + x2

14. 16c2 + 16c + 4

35

Lesson 5.4

Factor the difference of two squares:

15. x2 - 16

16. r2 - 100

17. x2 - 81

18. m2 - 1

19. 49 - x2

20. x2 - 49

21. 4m2 - 25

22. 25r2 - 36

23. 4n2 - 81

24. 144 - 25x2

25. 1 - 64p2

26. .16 - 25n2

27. 1.44r2 - 1.21t2

28. 169x2 - 225

29. 16t2 - 625

30. 49m2 - 256

36

Lesson 5.5 Factoring Combined Types Unit 5

Rules: In order to factor some polynomials completely, it may be

necessary to use more than one type of factoring.

To factor a polynomial completely: First, factor out the GCF

other than 1, if any exists. Then factor the remaining

polynomial, if possible, until it factors no further.

Examples: 1. Factor completely: 3x2 - 9x - 30

3x2 - 9x - 30 . Look for common factors.

3(x2 - 3x - 10) . 3 is the GCF.

3(x )(x ) . Factor x2 - 3x - 10 into two binomials,

if possible.

3(x - 5)(x + 2) . Try combinations of factors of -10:

-5 and + 2 are correct.

- 5x

+2x . -5x + 2x = -3x, the correct term.

So, the complete factorization of 3x2 - 4x - 30 is

3(x - 5)(x + 2).

2. Factor completely: 30a3 + 22a2 - 28a

30a3 + 22a2 - 28a . Look for common factors.

2a(15a2 + 11a - 14) . 2a is the GCF.

2a( )( ) . Factor the trinomial into two

binomials, if possible.

2a(5a )(3a ) . Try 5a and 3a as factors.

2a(5a + 7)(3a - 2) . Try combinations of factors of -14:

+7 and -2 are correct.

+ 21a

- 10a . 21a - 10a = 11a, the correct term.

So, the complete factorization of 30a3 - 22a2 - 28a is

2a(5a + 7)(3a - 2).

Practice: A. Factor completely: 5r2 - 35r + 50

37

Lesson 5.5

Examples: 3. Factor completely: 5x3 - 125x

5x3 - 125x . Look for common factors.

5x(x2 - 25) . 5x is the GCF.

5x( + )( - ) . x2 - 25 factors into a sum and a difference.

5x(x + 5)(x - 5) . Completely factored form.

4. Factor completely: 162 - 2y4

162 - 2y4 . Look for common factors.

2(81 - y4) . 2 is the GCF.

2[92 - (y2)2] . 81 - y4 is a difference of two squares:

y4 is the square of y2.

2(9 + y2)(9 - y2) . A difference of two squares factors

into a sum and a difference.

2(9 + y2)(3 + y)(3 - y) . 9 - y2 is a difference of two squares,

but 9 + y2 is not.

So, the complete factorization of 162 - 2y4 is

2(9 + y2)(3 + y)(3 - y).

HOMEWORK Factor completely, if possible:

1. 3x2 + 15x + 18 11. 2p2 - 4p

2. 2a2 + 6a - 8 12. 6m3 - 3m2 - 30m

3. 5r2 - 80 13. 2a3 + 26a2 - 6a

4. 2m3 - 4m2 - 6m 14. 96a3 - 6a

5. 2c3 - 3c2 - c 15. 4m2 + 7m - 2

6. 4x3 - 9x 16. 2k2 - 7k - 15

7. 2x3 - 2x2 - 40x 17. 5a3 - 20a

8. 3a3 + 15a2 + 12a 18. 30x3 - 9x2 - 12x

9. x2 - 5x - 24 19. 12n3 - 84n2 + 39n

10. 4k3 - 16k2 + 12k 20. 625r4 - 16

38

Lesson 5.6 Factoring by Grouping Unit 5

Rule: Some polynomials may be factored by grouping two (or more)

terms having a common factor and then factoring out all

common factors, first from each group and then from the

whole polynomial.

Examples: 1. Factor:

2x + px . x is a common factor.

(2 + p)x . Factored form; common factor on

right.

2. Factor:

2(x + 1) + p(x + 1) . (x + 1) is a common factor.

(2 + p)(x + 1) . Factored form; common factor on

right.

3. Factor:

a(x + 1) - b(x + 1) . (x + 1) is a common factor.

(a - b)(x + 1) . Factored form; common factor on

right.

4. Factor:

(2x + 3)3m + (2x + 3)5r . (2x + 3) is a common factor.

(2x + 3)(3m + 5r) . Factored form; common factor on left.

5. Factor:

(3x + 2y)2a - 7b(3x + 2y) . (3x + 2y) is a common factor.

(3x + 2y)(2a - 7b) . Factored form; choose either side for

common factor.

39

Lesson 5.6

Practice: Factor out the common factor:

A. 5(r - 3) + m(r - 3)

B. 2a(x + 7) - b(x + 7)

Examples: 6. Factor: 2x + 2 + px + p

(2x + 2) + (px + p) . Group so terms in each group have a

common factor: 2 and p.

2(x + 1) + p(x + 1) . Factor the GCF from each group; (x + 1)

is now a common factor.

(2 + p)(x + 1) . (x + 1) is factored out; compare to

Example 2 above.

7. Factor: 6mx + 10rx + 9m + 15r

(6mx + 10rx) + (9m + 15r) . Group terms with common factors.

2x(3m + 5r) + 3(3m + 5r) . Factor each group.

(2x + 3)(3m + 5r) . Factored form; (3m + 5r) is a common

factor.

8. Factor: 6ax + 4ay - 21bx - 14by

(6ax + 4ay) + (-21bx - 14by) . Show addition of the groups when

third term is negative.

2a(3x + 2y) - 7b(3x + 2y) . Negative common factor (-7b) in

second group.

(2a - 7b)(3x + 2y) . Factored form; common factor is

(3x + 2y).

Here's another way to factor 6ax + 4ay - 21bx - 14by:

(6ax - 21bx) + (4ay - 14by) . Rearrange so third term is positive;

then group.

3x(2a - 7b) + 2y(2a - 7b) . Extract common factors from each

group.

(2a - 7b)(3x + 2y) . Factored form; common factor is

(2a - 7b).

40

Lesson 5.6

Practice: Factor:

C. 7ax + 3ay - 7bx - 3by

Rule: After factorization by grouping, be sure to check each

resulting factor for any possible further factorization.

Example: 9. Factor: 4a2m + 4a2 - 25m - 25

(4a2m + 4a2) + (-25m - 25) . Write as a sum of two groups

(binomials).

4a2(m + 1) - 25(m + 1) . Common factors are 4a2 and -25.

(4a2 - 25)(m + 1) . (m + 1) is a common factor.

(2a + 5)(2a - 5)(m + 1) . 4a2 - 25 is a difference of two squares.

Since no factor will factor further, the complete factorization of

4a2m + 4a2 - 25m - 25 is

(2a + 5)(2a - 5)(m + 1).

41

Lesson 5.6

HOMEWORK Use grouping techniques to factor each polynomial completely, if possible:

1. (c + d)x + (c + d)y

2. 2(m - n) + p(m - n)

3. (x + 5)a2 + 3(x + 5)

4. (c2 - 25)9 - d2(c2 - 25)

5. k(64 - a2) + (64 - a2)m

6. 81(r2 - 9) - m4(r2 - 9)

7. ak + am + bk + bm

8. xc - xd + 4c - 4d

9. xy + 2x + ry + 2r

10. x2 + 5x + xy + 5y

11. r2 - 5r + 3r - 15

12. c2 - 2c + cd - 2d

13. 2am + 5ak + 2bm + 5bk

14. cx + 6c + x2 + 6x

15. m2 + 3m - km - 3k

16. r2c + r2d - 9c - 9d

17. 25x + 25 - g2x - g2

18. m3 - 4mn2 + 2m2 - 8n2

19. x2t + 2x2 - 100t - 200

20. b4 - 81a2b2 - 4b2 + 324a2

42

Lesson 5.7 Solving Quadratic Equations by Factoring Unit 5

Rules: A quadratic equation is an equation of the form

ax2 + bx + c = 0. (Notice that one member is a second degree

polynomial and the other member is 0.)

Zero Product Property: If x . y = 0, then either x = 0, y = 0, or

both x and y are 0.

If the polynomial member can be factored, then the quadratic

equation may be solved by first factoring it and then applying

the Zero Product Property.

Example: 1. Solve: x(x + 2) = 0 . The product equals 0.

x = 0 or x + 2 = 0 . Set each factor equal to 0.

x = 0 or x = -2 . Solve each equation for x.

Check: x(x + 2) = 0 x(x + 2) = 0

Let x = 0. Let x = -2. 0(0 + 2) = 0 -2(-2 + 2) = 0

0(2) = 0 -2(0) = 0

0 = 0 0 = 0

TRUE, so 0 is a solution. TRUE, so -2 is a solution.

The solutions of x(x + 2) = 0 are 0 and -2.

Rule: For each quadratic, or second degree equation, two roots, or

solutions, must be found.

Example: 2. Solve: x2 - 12x + 35 = 0

(x - 7)(x - 5) = 0 . Factor.

x - 7= 0 or x - 5 = 0 . Set each factor equal to 0.

x= 7 or x = 5 . Solve each equation.

Check: x2 - 12x + 35 = 0 x2 - 12x + 35 = 0 .Try 7.

72 - 12(7) + 35= 0 52 - 12(5) + 35 = 0 . Try 5. 49 - 84 + 35= 0 25 - 60 + 3 = 0

0= 0 0= 0

TRUE TRUE

So, solutions are 7 and 5.

Practice: A. Solve: x2 - 5x - 14 = 0

43

Lesson 5.7

Note: Sometimes a double root or multiple root occurs in

a quadratic, cubic, or higher degree equation. The degree of an equation tells the maximum number of roots of the equation.

Example: 3. Solve: 4x3 - 40x2 + 100x = 0 . Third degree (cubic) equation.

4x(x2 - 10x + 25) = 0 . GCF is 4x.

4x(x - 5)(x - 5) = 0 . Completely factored form.

4x = 0 or x - 5 = 0 or x - 5 = 0 . 4 cannot equal 0.

x = 0 or x = 5 or x = 5 . Solve each equation and check.

Solutions are 0 and 5. Notice that 5 appears twice as a root.

For the cubic equation, 4x3 - 40x + 100 = 0, the three roots

are: 0 and the double root, 5.

Practice: B. Solve: 6x3 + 16x2 - 6x = 0

Example: 4. Solve: 15x2 + 14x - 8 = 0

(3x + 4)(5x - 2) = 0 . Factor the trinomial.

3x + 4 = 0 or 5x - 2 = 0 . Let each factor equal 0.

3x = -4 or 5x = 2 . Solve each equation.

x = -4/3 or x = 2/5 . Solutions are rational.

Both roots satisfy the original equation, so the solutions to

15x2 + 14x - 8 = 0 are -4/3 and 2/5.

Rule: Generally, a quadratic equation is easier to solve if it is first

written in standard form. A quadratic equation is in standard form when written: ax2 + bx + c = 0, where:

1. The polynomial equals 0.

2. Terms of the polynomial are arranged in descending

powers of the variable.

3. a, b, and c, are integers, with "a" positive.

Example: 5. Write 3n = 6 - 4n2 in standard form.

Since the n2 term is negative (-4n2), make the right member 0.

4n2 + 3n = 6 . Add 4n2 to each side.

4n2 + 3n - 6 = 0 . Subtract 6 on each side to

obtain standard form.

44

Lesson 5.7

Practice: Write each equation in standard form:

C. -5x2 + 2 = 4x

D. 3 = 4a - 2a2

E. 3m = 5m2 - 6

Example: 6. Solve: 5x = 6 + x2

Since the x2 term is positive, make the left member 0.

0 = 6 + x2 - 5x . Subtract 5x from each side.

0 = x2 - 5x + 6 . Arrange in descending powers of x.

0 = (x - 3)(x - 2) . Factor.

x - 3 = 0 or x - 2 = 0 . Let each factor be 0.

x = 3 or x = 2 . Solve each equation.

Check: 5x = 6 + x2 5x = 6 + x2

Try 3. 5(3) = 6 + (3)2 5(2) = 6 + (2)2 Try 2. 15 = 6 + 9 10 = 6 + 4

15 = 15; TRUE 10 = 10; TRUE

So, the solutions are 3 and 2.

Practice: Write in standard form; then solve each:

F. -2 = x2 - 3x G. 7 - x2 = 6x

Example: 7. Solve: 4 - 9x = -2x2

2x2 - 9x + 4 = 0 . Add 2x2 to each side and write

in standard form.

(2x - 1)(x - 4) = 0 . Factor.

2x - 1 = 0 or x - 4 = 0 . Let each factor be 0.

2x = 1 or x = 4 . Solve each equation.

x = 1/2 or x = 4 . Check; both equations are true.

So, the solutions are 1/2 and 4.

Rule: Some word problems result in a quadratic equation. The

same four-step method for problem-solving is used to solve

such word problems.

45

Lesson 5.7

Example: 8. Eight less than 6 times a number is equal to the square of

the number. Find the number:

8 less than 6x is x2 . Choose a variable.

Let x be the number.

6x - 8 = x2 . Plan and write an equation.

6x - 8 = x2 . Solve.

0 = x2 - 6x + 8 . Write in standard form.

0 = (x - 4)(x - 2) . Factor.

x - 4 = 0 or x - 2 = 0 . Let each factor be 0.

x = 4 or x = 2 . Solve each equation.

Check both roots in the equation. Then check them with the

conditions of the original problem. The roots of this equation

satisfy the conditions stated, so 4 and 2 are both solutions to

the problem.

HOMEWORK

Use the zero product property and solve each equation by factoring:

1. a2 - 4a = 0 7. 3r2 - 7r = 0

2. x2 + 8x - 9 = 0 8. 2c2 + 13c - 24 = 0

3. 3t2 - 13t - 10 = 0 9. 2m2 + 5m - 42 = 0

4. 3x2 - 22x + 7 = 0 10. 2x2 + 23x + 56 = 0

5. 2a2 - 11a + 5 = 0 11. 5r3 - 245r = 0

6. 4x2 - 400 = 0 12. g3 - g2 - 20g = 0

Solve each equation:

13. x2 + 8 = -6x 18. 8y + y2 + 15 = 0

14. -28 + x2 = -3x 19. 8x = 21 - 5x2

15. x2 = 9x 20. 9x + x2 = -20

16. -2x2 = -3 + 5x 21. 4m = -m2 - 4

17. 7x - 6x2 = -5 22. 17x - 14 = -6x2

Write an equation; then solve each problem:

23. Five times a number, decreased by 6, is the same as the square of the

number. Find the number.

24. The sum of 6 times a number and the square of the number is 16.

Find the number.

25. Twelve less than 9 times a number is the same as the square of the

number increased by two. Find the number.

46

Lesson 5.8 Consecutive Integer Problems Unit 5

Rule: Consecutive integers are obtained by starting with a given

integer and counting by one's.

Examples: List four consecutive integers beginning with:

1. 3 : 3, 4, 5, 6

2. -2: -2, -1, 0, 1

3. x: x, x+1, x+2, x+3

Practice: List four consecutive integers beginning with:

A. 19 :

B. -7 :

C. m :

Rule: Even integers are divisible by 2. Consecutive even integers are

obtained by starting with an even integer and counting by

two's. Consecutive odd integers are not divisible by 2 but are

also obtained by counting by two's starting with an odd

integer.

Examples: List four consecutive even integers beginning with:

4. 6: 6, 8, 10, 12 . Add 2 each time.

5. -4: -4, -2, 0, 2 . Notice: Zero is even.

6. x: x, x+2, x+4, x+6 . If x is even.

List four consecutive odd integers beginning with:

7. 11 : 11, 13, 15, 17 . Add 2 each time.

8. -1 : -1, 1, 3, 5 . Add 2 each time.

9. x : x, x+2, x+4, x+6 . If x is odd.

47

Lesson 5.8

Practice: List:

D. 3 consecutive even integers, beginning with -28.

E. 3 consecutive odd integers, beginning with c.

F. 3 consecutive even integers, beginning with m.

Rule: A consecutive integer problem may be solved using the four

basic steps of problem solving.

Example: 10. Find three consecutive integers whose sum is 66.

Let x = the first . Choose a variable; then

Let x + 1 = the second represent the integers.

Let x + 2 = the third

x + x + 1 + x + 2 = 66 . Plan and write an equation..

3x + 3 = 66 . Solve the equation.

3x = 63

x = 21 . First integer.

x + 1 = 22 . Second integer.

x + 2 = 23 . Third integer.

21 + 22 + 23 = 66 . Check: The roots of the

equation satisfy the original problem.

So, the solution is: 21, 22, 23.

Rule: It is necessary to check the roots of an equation to see that

they satisfy the conditions of the original problem.

Consecutive integer problems may result in equations that

give two sets of roots; however, one, none, or both sets of roots

may be a solution to the problem.

48

Lesson 5.8

Examples: 11. Find two consecutive odd integers whose product is 195.

Let x = first . Choose a variable; then

Let x + 2 = second represent each integer.

x(x + 2) = 195 . Plan and write the equation.

x2 + 2x = 195 . Solve the equation; quadratic

may have 2 solutions.

x2 + 2x - 195 = 0 Rewrite in standard form.

(x - 13)(x + 15) = 0 Factor.

x - 13 = 0 or x + 15 = 0 Let each factor be 0.

x = 13 or x = -15 First integer: 13 or -15.

x + 2 = 15 or x + 2 = -13 Second integer: 15 or -13.

(13)(15) = 195 . Check both pairs of (-15)(-13)

(-15)(-13) = 195 consecutive odd integers.

So, there are two solutions: 13 and 15; -15 and -13.

12. Find three consecutive even integers such that the square

of the second decreased by 5 times the square of the first

is 16 less than twice the third.

Let x = first . Choose a variable; then

Let x + 2 = second represent each integer.

Let x + 4 = third

(x+2)2 - 5x2 = 2(x+4) - 16 . Plan and write the equation.

x2 + 4x + 4 - 5x2 = 2x + 8 - 16 . Solve the equation:

-4x2 + 4x + 4 = 2x - 8 Simplify each side.

-4x2 + 2x + 12 = 0 Add -2x and 8 to each side.

2x2 - x - 6 = 0 Divide each side by -2.

(2x + 3)(x - 2) = 0 Factor.

2x + 3 = 0 or x - 2 = 0 Let each factor be 0.

2x = -3 or x = 2 Solve each equation.

x = -3/2 or x = 2 -3/2 is not an integer.

. Check:

Try: x = 2 = First Only these three consecutive

x + 2 = 4 = Second even integers satisfy the

x + 4 = 6 = Third problem.

So, there is only one solution: 2, 4, and 6.

49

Lesson 5.8

HOMEWORK

Write an equation and solve each problem:

1. Find three consecutive integers whose sum is 21.

2. Find three consecutive integers whose sum is 0.

3. Find three consecutive even integers whose sum is -54.

4. Find three consecutive odd integers whose sum is 51.

5. Find five consecutive integers such that four times the third, increased

by 15, is 5 times the last.

6. Find four consecutive odd integers such that the first equals the

product of the second and fourth.

7. Find two consecutive even integers whose product is 120.

8. Find two consecutive odd integers such that the square of the second,

increased by the first is 88.

9. Find two consecutive even integers such that twice the square of the

second increased by the first is 188.

10. Find two consecutive integers such that the sum of their squares is

221.

11. Find four consecutive integers such that the difference of the squares of

the second and fourth is 36.

12. Find three consecutive odd integers such that the square of the third

decreased by the square of the first is 8 times the second.

13. Find three consecutive integers such that the square of the first is 15

less than the square of the second.

14. Find three consecutive even integers such that the product of the first

and third is two more than 5 times the second.

15. Find four consecutive odd integers such that the square of the sum of

the first and fourth equals the square of the sum of the second and

third.

 

UNIT 6

MULTIPLYING AND DIVIDING RATIONAL EXPRESSIONS

6.0 Review . . . . . . . . . . . . . . . . . . 52 6.1 Rational Expressions. . . . . . . . . . . . . . 54 6.2 Rational Expressions: Simplest Form. . . . . . . . . 58 6.3 Using -1 in Factoring . . . . . . . . . . . . . . 62 6.4 Dividing Powers . . . . . . . . . . . . . . . 66 6.5 Simplifying a Product . . . . . . . . . . . . . 70 6.6 Multiplying and Dividing . . . . . . . . . . . . 74 6.7 Dividing by a Monomial . . . . . . . . . . . . . 78 6.8 Dividing by a Binomial . . . . . . . . . . . . . 82

52

Lesson 6.0 Review Unit 6

Multiplying Binomials: FOIL Method - Multiply First, Outer, Inner, and Last terms.

last -3

first 6x2

(3x + 1) (2x - 3) = (3x + 1)(2x - 3) = 6x2 - 7x - 3

inner 2x

outer -9x

Multiplying Polynomials: Multiply each term of one by each term of the other.

(2x + 3)(4x2 + 2x + 1) = 8x3 + 4x2 + 2x + 12x2 + 6x + 3

= 8x3 + 16x2 + 8x + 3

Factoring: Quadratic Trinomials:

third term 3x2 - 14x + 15 = (3x - 5)(x - 3) . Both negative.

positive 3x2 + 14x + 15 = (3x + 5)(x + 3) . Both positive.

third 6x2 - 11x - 10 = (2x - 5)(3x + 2) . Largest product is term negative.

negative 6x2 + 11x - 10 = (2x + 5)(3x - 2) . Largest product is

positive.

Special Cases:

perfect a2 + 2ab + b2 = (a + b)2 . square of

trinomial a2 - 2ab + b2 = (a - b)2 a binomial

square

difference a2 - b2 = (a + b)(a - b) . binomial sum times

of two binomial difference

squares

By Grouping:

x2 + 3x - kx - 3k

(x2 + 3x) - (kx + 3k) . Group.

x(x + 3) - k(x + 3) . Common factors are x and k.

(x - k)(x + 3) . Common factor is (x + 3).

53

Lesson 6.0

Zero Product Property:

If x . y = 0, then x = 0, y = 0, or both x and y are 0.

Solving Quadratic Equations: Solve: -14x = 6 - 12x2

12x2 - 14x - 6 = 0 . Write in standard form.

2(3x + 1)(2x - 3) = 0 . Factor non-zero member.

3x + 1 = 0 or 2x - 3 = 0 . Set each variable factor equal to zero.

3x = -1 2x = 3 . Solve each linear equation formed.

x = -1 x = 3 . Check each solution.

3 2

Review Problems Multiply:

1. 5x(3x - 4) 4. (2x - 1)(2x - 6)

2. (x + 4)(x - 3) 5. (3x - 4)2

3. (2x + 3)(3x + 4) 6. (3x + 2)(2x2 - x + 3)

Factor completely:

7. x2 - x - 12 12. 4x2 - y2

8. x2 + 9x + 8 13. x2y2 - 36p2

9. m2 - 6m - 27 14. 25a2 - 10ab + b2

10. 3m2 - 14m + 15 15. x4 + 2x2 + 1

11. 6r2 + 13r - 15 16. 6mr + 4mt - 21pr - 14pt

Solve each equation:

17. 3x2 + 15x + 18 = 0

18. 2r2 - 50 = 0

19. 7x - 2 = -4x2

20. -9x2 = 12x - 30x3

54

Lesson 6.1 Rational Expressions Unit 6

Rule: A rational expression is of the form a where "a" and "b" are

polynomials. b

A rational expression a is undefined (or meaningless) for b

any value of the variable which makes b = 0.

Example: 1. Recall 16 = 8, a unique number,

2 because only 8 . 2 = 16.

But 16 = no unique number because

0 (no number) . (0) = 16.

And 0 = no unique number because

0 (any number) . (0) = 0.

Rule: To find the value(s) of the variable for which a rational

expression is undefined, set the denominator equal to 0 and

solve the resulting equation.

Examples: 2. For what value of x is x - 4 undefined?

3x - 6

3x - 6 = 0 . Set the denominator equal to 0

3x = 6 and solve for x.

x = 2 . This value of x makes the

denominator equal to 0.

Thus, the fraction is undefined when x = 2.

55

Lesson 6.1

3. For what values of c is 2c + 5 undefined?

2c2 + 5c - 12

2c2 + 5c - 12 = 0 . Set the denominator equal to 0 and

solve for c.

(2c - 3)(c + 4) = 0 . Factor.

2c - 3 = 0 or c + 4 = 0 . Let each factor equal 0.

2c = 3 or c = -4 . Solve each equation.

c = 3 or c = -4 . These values of c make the

2 denominator equal 0.

So the fraction is undefined if c = 3 or if c = -4.

2

Practice: For what value(s) of the variable is each fraction undefined?

A. 4x2 - 25 B. 2m - 8

6x + 24 6m2 + 7m - 5

Rule: Multiply two rational expressions the same way rational

numbers are multiplied: First multiply numerators together

and then multiply denominators:

a . c = ac if b ≠ 0 and d ≠ 0

b d bd

56

Lesson 6.1

Examples: 4. Multiply:

3x2 . 7x2 =

2y 8y2

3x2 . 7x2 = . Numerator product.

2y . 8y2 . Denominator product.

21x4 . Multiply in numerator

16y3 and denominator.

5. Multiply:

_ m + 2 . 3m - 1 =

m - 3 2m - 3

_ (m + 2)(3m - 1) = . Show products in numerator

(m - 3)(2m - 3) and denominator.

_ 3m2 + 5m - 2 . Multiply in numerator

2m2 - 9m + 9 and denominator.

6. Multiply:

(x - 3) . (2x + 1) =

x - 5

x - 3 . 2x + 1 = . Write x - 3 in rational form: 1

x - 5 a/b, with b = 1.

(x - 3)(2x + 1) = . Numerator product.

(1) (x - 5) . Denominator product.

2x2 - 5x - 3 . Multiply in numerator

x - 5 and in denominator.

57

Lesson 6.1

HOMEWORK

Find the value(s) of the variable which make(s) each rational expression undefined:

1. 5x 5. 2m + 7

x - 3 3m2 - m - 2

2. x + 1 6. 2c

3x - 12 8c2 - 12c

3. 5a + 1 7. c2 - 4

3a - 15 16c2 - 81

4. 2a 8. 2r + 6

a2 - 9 r3 - 2r2 - 15r

Multiply:

9. 2a2 . 7a5 14. x - 4 . 3x + 4

5b 3b4 x x + 5

10. _ 3x3 . 9x 15. (m + 3) . 3m - 1

2y4 7y2 2m + 1

11. 2r5 . -13r5 16. (r + 1) . 3m - 1

7m3 5m2 r + 4

12. x - 1 . 2x - 3 17. x + 2y . x - 2y

x + 3 2x + 1 a - 3b a + 3b

13. _ a + 3 . 3a - 1 18. _ 2r - m . r - 3m

a - 4 2a + 1 4r + m r + 2m

58

Lesson 6.2 Rational Expressions: Simplest Form Unit 6 Rules: Two polynomials are relatively prime if their GCF is 1. A

rational expression is in simple form (reduced to lowest terms)

if its numerator and denominator are relatively prime.

Simplify a rational expression just as rational numbers are

simplified: First, factor numerator and denominator. Then

divide out all common factors.

Examples: 1. Simplify:

24x =

16x

8 . 3 . x = . Factor; 8 is common to the

8 . 2 . x numerator and denominator.

18 . 3 . x1 = . Divide out all common factors

18 . 2 . x1 (cancel).

3 . Multiply remaining factors.

2

2. Simplify:

x2 - 7x + 12 =

2x2 - 7x - 4

(x - 3) (x - 4) = . Factor numerator and denominator.

(2x + 1)(x - 4)

(x - 3) (x - 4) 1 = . Divide out common factors (cancel).

(2x + 7)(x - 4)1

x - 3 . Multiply remaining factors.

2x + 7

59

Lesson 6.2

3. Simplify:

r - 2 =

5r - 10

r - 2 = . Factor 5r - 10.

5(r - 2)

r - 21 = 1 . Cancel; then multiply remaining

5(r - 2)1 5 factors.

4. Simplify:

-4a3 + 100a =

a3 + 3a2 - 10a

-4a(a2 - 25) = . 4 and a are common.

a(a2 + 3a - 10) . a is common.

-4a1(a + 5)1(a - 5) = . Completely factor.

a1(a + 5)1 (a - 2)

-4a(a + 5)1(a - 5) = . Divide out common factors (cancel).

a(a + 5)1(a - 2)

-4(a - 5) or -4a + 20 . Multiply remaining factors.

a - 2 a - 2

5. Simplify:

x2 + 5x + 6 =

x2 - 1

(x + 2)(x + 3) = . Factor numerator and denominator.

(x - 1)(x + 1)

(x + 2)(x + 3) = . No factor is common to numerator

(x - 1)(x + 1) and denominator.

So, x2 + 5x + 6 is in simplest form.

x2 - 1

60

Lesson 6.2

HOMEWORK

Simplify, if possible:

1. 14m

7m

2. _ 20r

36

3. a - 4

3a - 12

4. x2 - 36

x2 - 8x + 12

5. _ k2 - 10k + 21

k2 + 3k - 28

6. _ m - 4

2m2 - 8m

7. x2 - 8x + 16

4x - 16

8. 3m + 4

6m + 4

9. c2 - 8c + 7

3c - 21

10. b2 - 25

b2 + 10b + 25

61

Lesson 6.2

11. x2 - 7x - 18

x2 + 2x

12. r2 - 4

2r2 + 3r - 2

13. 3x2 - 10x - 8

x2 + 4x

14. 2x2 - 11x + 12

4x2 - 4x - 3

15. 6d2 + 7d - 3

10d2 + 13d - 3

16. 3a2 - 16a - 12

9a2 + 12a + 4

17. 2n4 - 7n3 - 30n2

6n6 + 13n3 - 5n2

18. 6m4 - 7m3 - 5m2

3m4 + 7m3 - 20m2

19. cx - c2 - x + c

c2x2 - x2 - 9c2 + 9

20. a4 - 13a2 + 36

a3 + a2 - 6a

62

Lesson 6.3 Using -1 in Factoring Unit 6

Rule: A polynomial is in practical form when it is written in

descending order of exponents with the first, or leading,

coefficient positive. It is sometimes necessary to factor -1 from

a polynomial to obtain a positive leading coefficient.

Examples: Write each polynomial in practical form:

1. 3x + 2x2 - 5 . Rewrite in descending order of

2x2 + 3x - 5 exponents.

2. -6 + 5x - 7x2 . Rewrite in descending order of

-7x2 + 5x - 6 exponents.

-1(7x2 - 5x + 6) . Factor out -1. The leading

coefficient, 7 , is now positive.

Rule: A rational expression is easier to simplify if all polynomials

involved are in practical form.

Examples: 3. Simplify:

x - 3 =

9 - x2

x - 3 = . Rewrite denominator in

-1x2 + 9 descending order of exponents.

x - 3 = . Factor -1 from denominator.

-1(x2 - 9)

x - 3 = . Factor.

-1(x + 3)(x - 3)

x - 31 = . Divide out common factors.

-1(x + 3)(x - 3)1

1 =

-1(x + 3)

-1 or _ 1 . Either is standard form for a

x + 3 x + 3 fraction.

63

Lesson 6.3

4. Simplify:

m2 + 6m - 16 =

10 - 3m - m2

m2 + 6m - 16 = . Rewrite denominator in descending

-1m2 - 3m + 10 order of exponents.

m2 + 6m - 1 = . Factor out -1.

-1(m2 + 3m - 10)

(m + 8)(m - 2)1 = . Factor numerator and denominator;

-1(m + 5)(m - 2)1 then cancel.

m + 8 or _ m + 8 . Answer.

-1(m + 5) m + 5

5. Simplify:

-2x - x2 + 24 =

8 - 2x

-x2 - 2x + 24 = . Write in descending order.

-2x + 8

-1(x2 + 2x - 24) = . Factor out -1 from numerator

-1(2x - 8) and denominator.

-11(x + 6)(x - 4)1 = . Factor; divide out common factors.

-11(2)(x - 4)1

x + 6 . Answer.

2

64

Lesson 6.3

HOMEWORK

Simplify:

1. r - 3

9 - r2

2. 3 - x

5x - 15

3. a2 - a - 12

4 - a

4. 12 - 2c

c2 - 4c - 12

5. x2 - 6x + 9

6 - 2x

6. m2 - m - 20

5 - m

7. -n2 + 5n - 4

n - 4

8. a2 - a - 42

14 + 5a - a2

9. -c2 + 12c - 20

c2 - 8c - 20

10. 4p - p2

-p2 + 5p + 14

65

Lesson 6.3

11. -r2 + 2r + 8

-2r2 - r + 6

12. 21 + 11t - 2t2

t2 - 2t - 35

13. -x2 + 5x + 24

21 - x2 + 4x

14. 15 + 2a - a2

2a2 + 20a + 42

15. 21 - 18x - 3x2

-4x + 28

16. 24 - 2m - 2m2

2m2 - 6m

17. 8r2 - 8t2

-4r2 + 20rt - 16t2

18. -20 + 44p - 4p2

3p2 - 33p + 15

19. 81 - m4

36 - 5m2 - m4

20. -x2 - 4xy + 21y2

x2 + 7xy - 3x - 21y

66

Lesson 6.4 Dividing Powers Unit 6

Rule: Rational expressions involving powers greater than 1 can be

simplified by expanding powers and then "cancelling" (or

dividing out) all factors common to numerator and

denominator. Remaining like factors may then be multiplied.

Examples: Simplify:

1. x6 = x . x . x . x . x . x . One method is to expand

x4 x . x . x . x powers.

= x1 . x1 . x1 . x1 . x . x . Cancel: divide in numerator

x1 . x1 . x1 . x1 and denominator by 4 factors of x.

= x . x = x2 . Two factors of x remain in

1 numerator.

2. c3 = c . c . c . Expand powers.

c5 c . c . c . c . c

= c1 . c1 . c1 . Cancel: divide numerator and

c1 . c1 . c1 . c . c denominator by 3 factors of c.

= 1 = 1 . Two factors of c remain in the

c . c c2 denominator.

Practice: Expand powers and cancel like factors to simplify:

A. a7 = B. x7 =

a4 x9

67

Lesson 6.4

Rule: Rational expressions are sometimes simplified more easily by

using the Quotient of Powers Property:

If m > n, xm = xm-n, or xm-n

xn 1

If m < n, xm = 1

xn xn-m

Examples: Simplify:

3. a9 = a9-4 = a5 = a5 . m > n.

a4 1 1

4. c4 = 1 = 1 . m < n.

c9 c9-4 c5

5. x6y5 = x6 . y5 . Separate into a product

x9y4 x9 y4 of two fractions.

= 1 . y5-4 . Quotient of Powers

x9-6 1 Property used twice.

= 1 . y . Simplify exponents.

x3 1

= y . Multiply fractions.

x3

6. The Quotient of Powers Property is often applied mentally:

x6y5 = x6 . y5 . Think: x6 = 1 = 1

x9y4 x9 y4 x9 x9-6 x3

Also: y5 = y5-4 = y

y4 1 1

= 1 . y . Write the remaining factors.

x3 1

= y . Multiply the remaining

x3 factors.

68

Lesson 6.4

Practice: Use the Quotient of Powers Property to simplify:

C. a8b3 D. 14x13t5

a6b12 21x8t9

Rule: To simplify a rational expression, completely factor numerator

and denominator (factor out -1 where needed) and apply the

Quotient of Powers Property.

Example: 7. Simplify:

25(a3 - 4a2 + 3a) =

10a2(9 - a2)

25a(a2 - 4a + 3) = . "a" is a common factor.

10a2(-1)(a2 - 9) . Factor out -1.

525a1(a - 3)1(a - 1) = . Factor completely and

210a2(-1)(a-3)1(a+3) divide out common factors.

a 5(a - 1) =

2a(-1)(a+3)

_ 5(a - 1) . Answer.

2a(a+3)

69

HOMEWORK Simplify:

1. c4 7. 16x12r8

c9 24x3r3

2. x11 8. 3a6m3

x7 12a3m12

3. m5 9. 12gh3

m8 4g2h4

4. r3t5 10. 8a9c6

rt2 10a3c7

5. m5b9 11. 3x + 9

m8b6 x2 - 81

6. 6a4b7 12. 2x2 - 4x

b8c 4x2 - 64

Simplify:

13. 2a2(5a + 2)

50a3 - 8a

14. -7m2(3 - x)

21m(x2 - 9)

15. x3(6m - 30)

x9(2m2 - 50)

16. 2r3(72 - 2t2)

4r5(t2 - 12t + 36)

17. m5(1 - r)

m4r - m4

18. c2d(3a2 - 12a + 9)

cd2(6a - 18)

19. x2(24 - 6r)

3x4r2 - 48x4

20. 2a3 - 14a2 + 20a

60a2 + 8a3 - 4a4

70

Lesson 6.5 Simplifying a Product Unit 6

Rule: To simplify a product of two (or more) rational expressions,

write the product of their numerators divided by the product

of their denominators:

a . c = a . c (b ≠ 0; d≠ 0)

b d b . d

Then simplify the resulting rational expression by applying

one or more of the techniques learned in prior lessons.

Examples: Simplify each product:

1. a2 - 4 . a3 - 9a =

a + 3 a2 - 5a + 6

(a2 - 4)(a3 - 9a) = . Product of numerators.

(a + 3)(a2 - 5a + 6) . Product of denominators.

(a +2)(a-2)a(a+3)(a-3) = . Factor numerator and

(a+3)(a-2)(a-3) denominator.

(a+2)(a-2)1a(a+3)1(a-3)1 = . Divide out common factors.

1(a+3)1(a-2)(a-3)1

a(a + 2) or a(a + 2) . Answer.

1

71

Lesson 6.5

2. 3(8 - 2x) . -2x - 2 =

3x2 - 9x - 12

3(8 - 2x) . -2x - 2 = . Write 3(8-2x) in fraction form.

1 3x2 - 9x - 12

3(8 - 2x) . (-2x - 2) = . a . c = a . c

1 . (3x2 - 9x - 12) b d b . d

3(-2)(x - 4) . (-2)(x + 1) = . Factor the negative factor

1 . 3(x2 - 3x - 4) from each.

3(-2)(x - 4) . (-2)(x + 1) = . Completely factor.

1 . 3(x - 4)(x + 1)

31(-2)(x-4)1 . (-2)(x+1)1 = . Divide out common factors.

1 . 31(x - 4)1(x + 1)1

(-2)(-2) = 4 or 4 . Answer.

1 1

3. 5a3b2 . 8a2 - 48ab + 72b2`=

6b - 2a 20a2b3

5a3b2 . (8a2 - 48ab + 72b2) = . a . c = a . c

(6b - 2a) . 20a2b3 b d b . d

5a3b2 . 8(a2 - 6ab + 9b2) = . Factor GCF, 8.

-2(a - 3b) . 20a2b3 . Factor out -2.

5a3b2 . 8(a - 3b)(a - 3b = . Completely factor.

-2(a - 3b) . 20a2b3 a 1 2 1

15a3b2 . 8(a - 3b)1(a - 3b) = . Divide out common factors.

-2(a - 3b)1 . 20a2b3

-1 4 1 • 1 • b

a(a - 3b) or _ a(a - 3b) . Answer.

-b b

72

Lesson 6.5

HOMEWORK

Simplify each product:

1. x2 + 7x + 10 . 2x + 6

x + 3 x + 5

2. 5a + 20 . a + 2

a2 - 5a - 14 a + 4

3. (m - 2) . 4m - 12

m2 - 5m - 6

4. 6r + 36 . (r - 3)

3r2 - 27

5. 6a7x5 . 25

10 18a5x7

6. 5m5r2 . 8

48 25m4r5

7. x2 - 36 . 6

48 x2 - x - 30

8. 21 - 7a . 2a3b6

21a4b2 a2 + 5a - 24

9. x2 - x - 30 . 12x2y3

8xy2 12 - 2x

10. a2 + 6a - 27 . 7 - a

a2 - 49 a + 9

73

Lesson 6.5

11. m2 + m - 6 . 2m2 - 4m - 16

m2 - 6m + 8 m2 - 6m + 8

12. 5 + 4x - x2 . x2 - 5x + 6

x2 - 7x + 10 3 + 2x - x2

13. r2 + 3r . r2 + 4r - 21

r + 7 r3 - 9r

14. a2 - a - 6 . a2 - 6a - 16

a2 - 3a - 40 a2 - 2a - 8

15. t2 - 10t + 16 . 2t2 + 11t - 21

2t2 - 3t t2 - t - 56

16. m2 - 7mn - 18n2 . 3m - 6n

- m + 9n m2 - 4n2

17. 3r2 - 10r - 8 . 2r2 + r - 15

2r2 - 13r + 20 -3 - r

18. 25x2 - m2 . 2x2+9xm-56m2

5x2+41xm+8m2 2x2+3xm-35m2

19. c2 - 1 . 4c2-20c+21 . c2+8c+7

2c2-5c-7 4c2-12c+9 21-4c-c2

20. cx - 2cy - 2x +4y . x2 - 7xy + 6y2

x2 - y2 cx - 6cy - 2x + 12y

74

Lesson 6.6 Multiplying and Dividing Unit 6

Rule: The reciprocal (or multiplicative inverse) of a rational expression

a, where a ≠ 0, is b.

b a

The product of any quantity and its reciprocal is 1:

a . b = 1 or b . a = 1

b a a b

Examples: Write the reciprocal of each number or expression:

Expression Reciprocal

1. 3 4

4 3

2. 5 1

5

3. 2 x + 5

x + 5 2

4. 0 None . No answer multiplied by 0 gives a

product of 1. (1/0 is undefined.)

Rule: To divide by a rational expression, multiply by its reciprocal:

If b ≠ 0, c ≠ 0 and d ≠ 0, a ÷ c = a . d

b d b c

Examples: 5. Simplify:

x2 - 12x + 32 ÷ 5x - 20 =

x2 - 25 x - 5

x2 - 12x + 32 . x - 5 = . Change to a product.

x2 - 25 5x - 20

(x2 - 12x + 32)(x - 5) = . Multiply.

(x2 - 25)(5x - 20)

(x - 4)1(x - 8)(x - 5)1 = . Factor and divide out

(x + 5)(x - 5)1(5)(x - 4)1 common factors.

x - 8 . Answer.

5(x + 5)

75

Lesson 6.6

6. Simplify:

12c7d2 ÷ 3c5d3 =

18c - 9c2 3c2 + 21c - 54

12c7d2 . 3c2 +21c - 54 = . a ÷ c = a . d

18c - 9c2 3c5d3 b d b c

12c7d2(3c2 +21c - 54) = . Multiply.

-1(9c2 - 18c)(3)c5d3 1 1

412c7d2(3)1(c - 2)1(c + 9) = . Factor and cancel.

(-1)9c1(c - 2)1(3)1c5d3d

3

_ 4c(c + 9) . Answer.

3d

Rule: When an expression includes both multiplication and division

of rational expressions, follow the order of operations.

Example: 7. Simplify:

t ÷ 3t2 . 9t =

t + 3 4t2 - 36 t2 - 5t + 6

t . 4t2 - 36 . 9t =

t + 3 3t2 t2 - 5t + 6

t . 4(t + 3)(t - 3) . 9t =

t + 3 3t2 (t - 3)(t -2)

t1 . 4(t + 3)1(t - 3)1 . 39t1 =

t + 31 3t2 (t - 3)1(t - 2)

t 1

12 . Answer.

t - 2

76

Lesson 6.6

HOMEWORK

Simplify:

1. x2 - 3x - 10 ÷ 5x - 25

x2 - 25 7x + 35

2. 3m - 18 ÷ m2 - 36

6 2m + 12

3. k2 - 4 ÷ 3k + 6

r + 1 9r + 9

4. 9c8 ÷ 5c4

3c2 - 15c c2 - 3c - 10

5. x2 + 2x + 1 ÷ x2 - 1

7x5 56x2

6. m4c ÷ 3m3c5

5m - 25 15m - 3m2

7. 36 - a2 ÷ 3a2 - 18a

ad6 27a2d5

8. x2 + 2x + 1 ÷ x2 - 1

2x - 6 x2 - 4x + 3

9. r - 3 ÷ r2 - 6r + 9

1 r

10. 3k2 + 17k + 20 ÷ 3k2 + 5k

6k5c7 4kc3

77

Lesson 6.6

11. 8m5p3 ÷ 4m9p5

5m2 - 10m 5m2 - 16m - 52

12. a2 + 7a + 10 . 6a2 ÷ 3a + 6

9a a + 1 a2 + a

13. 2x2 ÷ x2 - 5x . x2 - 9x + 20

x + 4 4 2x3

14. 4rt3 ÷ m4 . 6r2t

7m - 35 m2 - 5m m3

15. a2 + 6a - 27 ÷ a2 + 5a - 36

a2 - 3a - 40 a2 + a - 20

16. 16r2 - 9t2 ÷ 4r2 + 11rt + 6t2

r2 + 2rt r2

17. 4c4 - 25c2 ÷ 4c3 - 16c2 - 20c

6c2 - 23c + 20 20 + c - 12c2

18. 3x + 9y ÷ 9x ÷ x2 - m2

x2 - mx - 2m2 3x + 15m x2 - 4mx - 5m2

19. a2 - 4a - 21 . 2a2 - 10a ÷ a2 - 12a + 35

16a - 4a2 a2 - 11a + 24 a2 + 4a -32

20. 2cx + 6c + 5x + 15 ÷ x2 - 9

2c + 5 3x - x2

78

Lesson 6.7 Dividing by a Monomial Unit 6

Rule: By the Quotient of Powers Property:

a5 = a5-3 = a2

a3

The problem may also be written:

a5 ÷ a3 = a2

This is true because a3 . a2 = a5.

Example: 1. Divide: 2. Divide:

x7 ÷ x4 = x9 ÷ x =

x7-4 = x9-1 =

x3 . Quotient x8 . Quotient

Practice: A. Divide: m11 ÷ m3 B. Divide: a3 ÷ a

Rule: Monomials with coefficients other than 1 can be divided in the

same manner.

Examples: 3. Divide:

20a5 ÷ 5a3 =

20a5 =

5a3

20 • a5 =

5 a3

4a5-3 =

4a2 . Check: 5a3(4a2) = 20a5.

4. -45a6 ÷ 5a5 =

-45 a6-5 =

5

- 9a . Check: 5a5 . (-9a) = -45a6.

79

Lesson 6.7

5. 35x3 =

7x5 1

535 . x3 = . Factor the quotient.

17 x5x2

5 . Answer.

x2

Rule: To divide a polynomial by a monomial, divide each term of the

polynomial by the monomial.

Example: 6. Divide:

12a6 - 18a5 + 6a4 =

3a3

12a6 _ 18a5 + 6a4 = . Think: Divide each term by 3a3.

3a3 3a3 3a3

4a3 - 6a2 + 2a . Answer: Simplify each term.

This quotient may also be shown like this:

4a3 - 6a2 + 2a

3a3 ) 12a6 - 18a5 + 6a4

Check: 3a3(4a3 - 6a2 + 2a) = . Distribute the 3a3.

3a3 . 4a3 - 3a3 . 6a2 + 3a3 . 2a =

12a6 - 18a5 + 6a4

7. Divide: (16x8 + 8x5 - 10x2) ÷ 2x2 =

8x6 8x6

2x2) 16x8 + 8x5 - 10x2 = . Think: 2x2) 16x8

8x6 + 4x3 4x3

2x2) 16x8 + 8x5 - 10x2 = . Think: 2x2) 8x5

8x6 + 4x3 - 5 -5 2x2) 16x8 + 8x5 - 10x2 . Think: 2x2) -10x2

The quotient is 8x6 + 4x3 - 5.

80

Lesson 6.7

HOMEWORK

Divide:

1. n10 ÷ n3

2. m7 ÷ m5

3. x9 ÷ x8

4. x4 ÷ x

5. 18a5 ÷ 6a3

6. 33r3 ÷ (-11r)

7. -6a4 ÷ 3a

8. 12t4 ÷ 4t

9. -36p3 ÷ 4p3

10. 40m8 ÷ 5m5

11. -42x2 ÷ (-6x2)

12. 35a ÷ 5a

13. 48r ÷ 3r

14. 8x7 ÷ x

15. 45t3 ÷ 3t3

16. -35r5 ÷ (-5r4)

17. (12a3 - 8a2) ÷ 4a

18. (24x5 - 42x3) ÷ 6x3

81

Lesson 6.7

19. 3m3 - 15m

3m

20. 18c5 - 27c3

9c2

21. 24a3 - 32a2 + 12a

4a

22. 5d ) 35d5 - 15d3 - 20d

23. -4a5) 36a9 - 24a7 + 12a5

24. (12r5 +15r4 - 3r3) ÷ 3r3

25. (x8 - x5 + 3x3 + 7x2) ÷ x2

26. 8m5 + 4m4 - 16m3 - 24m2

-4m2

27. t) t8 - 5t7 + 4t6 - 8t

28. (-8r4 - 24r3 - 56r2 + 96r) ÷ (-8r)

29. 18f8 - 30f7 - 12f6 + 108f5

3f4

30. (a5r + a3r - a8r) ÷ ar

82

Lesson 6.8 Dividing by a Binomial Unit 6

Rule: To divide a polynomial by a binomial (or by another polynomial of

lower degree), follow the same procedure as in "long division":

1. Divide by a "trial divisor" (the first term).

2. Multiply divisor by quotient from Step 1.

3. Subtract this product from similar terms of the

polynomial.

4. "Bring down" the next term.

5. Repeat Steps 1-4 until no term is available to "bring

down."

Example: 1. Divide:

(x2 + 12x + 20) ÷ (x + 2)

x . Trial divisor is x.

x + 2) x2 + 12x + 20 Think: x) x2 = x.

x2 + 2x . Multiply: x(x + 2).

0x2 + 10x . Subtract. Then "bring down"

the next term, 20.

x + 10 . Now repeat the above steps.

x + 2) x2 + 12x + 20

x2 + 2x 10

10x + 20 . Divide: Think x) 10x.

10x + 20 . Multiply: 10(x + 2).

0x + 0 . Subtract.

The quotient is x + 10 with remainder 0.

83

Lesson 6.8

2. Divide:

(6n2 + 11n - 12) ÷ (3n - 2)

2n 2n 3n - 2) 6n2 + 11n - 12 . Divide: 3n) 6n2.

6n2 - 4n . Multiply: 2n(3n - 2).

0n2 + 15n . Subtract: 6n2 + 11n - (6n2 - 4n).

2n + 5

3n - 2) 6n2 + 11n - 12 . "Bring down" -12.

6n2 - 4n . Divide: 5

15n - 12 3n) 15n

15n - 10 . Multiply: 5(3n - 2).

0n - 2 . Subtract: 15n - 12 - (15n - 10).

The quotient is 2n + 5 with a remainder of -2.

Rule: When a polynomial dividend has a missing term, represent it with

a coefficient of "0" before performing the division.

Examples: 3. Divide:

(4r3 - 4r + 6) ÷ (2r - 1)

2r2 2r2 2r - 1) 4r3 + 0r2 - 3r + 6 . Divide: 2r)4r3.

4r3 - 2r2 . Multiply: 2r2(2r - 1).

2r2 . Subtract: 0r2 - (-2r2).

2r2 + r 2r - 1) 4r3 + 0r2 - 3r + 6 . "Bring down" 3r .

4r3 - 2r2 r

2r2 - 3r . Divide: 2r) 2r2

2r2 - r . Multiply: r(2r - 1).

- 2r . Subtract: -3r - (-r).

84

Lesson 6.8

2r2 + r - 1 2r - 1) 4r3 + 0r2 - 3r + 6 . "Bring down" 6 .

4r3 - 2r2

2r2 - 3r

2r2 - r -1

- 2r + 6 . Divide: 2r) -2r

- 2r + 1 . Multiply: -1(2r - 1).

5 . Subtract: 6 - 1.

Thus, the quotient is 2r2 + r - 1 with a remainder of 5.

4. Divide:

(9a3 - 4a + 2) ÷ (3a - 1)

Put all the steps together to make the division more

compact.

3a2 + a - 1 3a2

3a - 1) 9a3 + 0a2 - 4a + 2 . 3a) 9a3

9a3 - 3a2

3a2 - 4a a

3a2 - a . 3a)3a2

- 3a + 2

- 3a + 1 -1

1 . 3a) -3a

So, the quotient is 3a2 + a - 1 with a remainder of 1.

85

Lesson 6.8

HOMEWORK

Divide:

1. (x2 + 3x + 2) ÷ (x + 1)

2. (a2 - 5a + 6) ÷ (a - 2)

3. (m2 + 6m + 15) ÷ (m + 1)

4. (r2 - 9r + 20) ÷ (r - 4)

5. (p2 + 3p - 4) ÷ (p + 4)

6. (x2 + 8x - 20) ÷ (x - 2)

7. (2y2 - 6y - 216) ÷ (y - 12)

8. (12x2 + 46x + 40) ÷ (2x + 5)

9. (2a2 + 2a - 12) ÷ (a - 2)

10. (x2 + 11x + 24) ÷ (x + 3)

11. (2r2 - 14r + 6) ÷ (r - 2)

12. (15m2 + 8m + 1) ÷ (3m + 1)

13. (6x3 + 2x2 - 3x - 1) ÷ (2x2 - 1)

14. (3y4 + 27y2 + 60) ÷ (y2 + 4)

15. (k3 + 1) ÷ (k + 1)

16. (k3 + 1) ÷ (k - 2)

17. (4x3 - 18x2 + 22x - 6) ÷ (2x - 3)

18. (17x + 9x2 + 8 + 10x3) ÷ (2 + 5x)

19. (4t3 - 52t - 48) ÷ (2t + 6)

20. (x4 + 7x3 + 9x2 - 11x - 6) ÷ (x2 + 2x2)

 

iii

UNIT 7

ADDING AND SUBTRACTING RATIONAL EXPRESSIONS

7.0 Review . . . . . . . . . . . . . . . . . . . . 88

7.1 Adding Rational Expressions with Monomial Denominators. . . 90

7.2 Adding Rational Expressions with Polynomial Denominators. . 96 7.3 Adding Other Types of Rationals . . . . . . . . . . 100 7.4 Subtracting Rational Expression. . . . . . . . . . . 104 7.5 Simplifying by Factoring Out -1 . . . . . . . . . . . 108 7.6 Complex Rational Expressions . . . . . . . . . . . 112

7.7 Complex Rational Expressions: Polynomial Denominators . . 118

88

Lesson 7.0 Review Unit 7

Rational Expressions: a where "a" and "b" are polynomials but undefined for values that

b

make b = 0. Example: 2x + 1 is undefined only when x = 2.

3x - 6

Multiplying Rational Expressions: a . c = ac Example: x - 4 . 3x + 4 = (x-4)(3x + 4) = 3x2- 8x - 16

b d bd x x + 5 x(x + 5) x2 + 5x

Simplifying a Rational Expression (Reducing): 1. Factor numerator and denominator.

2. Divide out all common factors.

Example:

x2 + 3x - 4 = x2 + 3x - 4 = (x + 4)(x - 1)1 = _ x + 4

1 - x2 -(x2 - 1) (x + 1)(x - 1)1 x + 1

Quotient of Powers Property: If m > n, xm = xm-n, but if m < n, xm = 1

xn xn xn-m

Examples: 1. x8 = x8-5 = x3 2. x5 = 1 = 1

x5 x8 x8-5 x3

Dividing Rational Expressions: 1. Follow order of operations.

2. To divide by a rational expression, multiply by its reciprocal.

Example: x ÷ 3x2 . 9x =

x + 3 4x2 - 36 x2 - 5x + 6

x1 . 4(x + 3)1(x - 3)1 . 39x1 = 12

x + 31 3x2 (x - 2)(x - 3)1 x - 2

x 1 Dividing a Polynomial by a Monomial: Divide each term by the monomial.

1. 12m3 + 32m2 - 24m = 12m3 + 32m2 - 24m = 3m2 + 8m - 6

4m 4m 4m 4m or

3m2 + 8m - 6

2. 4m ) 12m3 + 32m2 - 24m

89

Lesson 7.0

Dividing a Polynomial by a Polynomial of Lower Degree: 1. Divide by "trial divisor" (first term).

2. Multiply divisor by quotient from Step 1.

3. Subtract this product.

4. "Bring down" the next term.

5. Repeat Steps 1-4.

Example:

5x + 8

3x - 2 ) 15x2 + 14x - 16

15x2 - 10x

24x - 16

24x - 16

Review Problems Find the value(s) of the variable for which each rational expression is undefined:

1. 5x + 1 2. 2x2 + x + 1

3x - 15 3x2 - x - 2

Simplify each rational expression:

3. x - 3 9. 14a13b5

5x - 15 21a8b9

4. a2 + 5a - 6 10. 2x3(72 - 2y2)

a2 - 1 4x5(y2 - 12y + 36)

5. 3m2 - 16m - 12 11. 5a5b2 . 25

9m2 + 12m + 4 48 18a6b7

6. x2 - x - 20 12. 3x - 18 ÷ x2 - 36

5 - x 6 2x + 12

7. -x2 + 12x - 20 13. 3a + 9m ÷ 9a ÷ a2 - m2

x2 - 8x - 20 a2-ma-2a2 3a+15m a2-4ma-5m2

8. 4a - b2

-b2 + 5b + 14

Divide:

14. 35x5 - 15x3 - 20x 15. 15x2 + 8x + 1

5x 3x + 1

90

Lesson 7.1 Adding Rational Expressions with Monomial Denominators Unit 7

Rule: The Addition Property for Rational Expressions says that for

all numbers a and c and each non-zero number b,

a + c = a + c

b b b

In the remainder of this lesson, no denominator will be

assumed to be zero.

Examples: 1. Add: 3 + 7 = 2. Add: 3m + 2m =

x x a - 4 a - 4

3 + 7 = 3m + 2m =

x a - 4

10 5m

x a - 4

Rule: Some rational expressions can be written in simpler form.

After adding, factor and divide out common factors.

Examples: 3. Add: 4. Add:

9a + 5a = t2 + 6t + 8 =

7 7 3t + 12 3t + 12

9a + 5a = t2 + 6t + 8 =

7 3t + 12

14a = (t + 2 ) (t + 4) =

7 3(t + 4)

214a = (t + 2) (t + 4)1 =

71 3(t + 4)1

t + 2

2a 3

Practice: C. Add: 8m + 4m D. Add: k2 + -4

3 3 8k - 16 8k - 16

Rule: The Addition Property can be extended to more than two

rational expressions:

a + b + c = a + b + c

d d d d

91

Lesson 7.1

Example: 5. Add: x2 - 7x + 11x - 8 + 3 =

x2 + 3x - 10 x2 + 3x - 10 x2 + 3x - 10

x2 - 7x + 11x - 8 + 3 =

x2 + 3x - 10

x2 + 4x - 5 =

x2 + 3x - 10

(x - 1) (x + 5) =

(x - 2) (x + 5)

(x - 1) (x + 5)1 = x - 1

(x - 2) (x + 5)1 x - 2

Rule: When both numerator and denominator of a rational

expression are multiplied by the same non-zero real number,

an equivalent rational expression is obtained:

For all numbers a, and non-zero b and c,

a = ac

b bc

Use this rule to add rationals with unlike denominators.

Example: 6. Add: 2x + 6x =

5 35

2x + 6x = . Factor denominators.

5 5 . 7

2x . 7 + 6x = . Determine LCD. Multiply by 7 in

5 . 7 5 . 7 numerator and denominator.

14x + 6x = . Multiply; common denominator is 35.

35 35

20x = . Addition property for rational

35 expressions.

4x . Answer.

7

92

Lesson 7.1

Rule: When adding rational expressions with unlike denominators,

every factor that appears in any denominator must be

included in the LCM of the denominators. This LCM is the

lowest common denominator (LCD) for the expressions.

Examples: 7. Add: m + 3m + 2 =

12 6 18

m + 3m + 2 = . At most, a denominator has

2 . 2 . 3 2 . 3 2 . 3 . 3 two 3's and two 2's; so, the LCD is 2 . 2 . 3 . 3.

m + 3m + 2 = . Compare each denominator to

2 . 2 . 3 2 . 3 2 . 3 . 3 LCD; find missing factors for each.

needs needs needs

3 2 . 3 2

m . 3 + 3m . 2 . 3 + 2 . 2 = . Multiply each

2 . 2 . 3 . 3 2 . 3 . 2 . 3 2 . 3 . 3 . 2 numerator and denominator by its missing

factors.

3m + 18m + 4 = . Multiply.

36 36 36

3m + 18m + 4 = . Combine numerators.

36

21m + 4 . Simplify the answer.

36

93

Lesson 7.1

8. Add:

2a + 3 + 4a - 1 =

6a 9a

2a + 3 + 4a - 1 = . Factor denominators; LCD is

2 . 3 . a 3 . 3 . a 2 . 3 . 3 . a.

needs 3 needs 2 . Find missing factors.

(2a + 3)3 + (4a - 1)2 = . Multiply numerator and

2 . 3 . a . 3 3 . 3 . a . 2 denominator by the missing factors.

6a + 9 + 8a - 2 = . Simplify numerators and

18a 18a denominators.

14a + 7 . Combine numerators for

18a simplified form.

9. Add:

x + 1 + 2x - 4 + 10 =

2x 3x 4

x + 1 + 2x - 4 + 10 = . Factor: LCD is 2 . 2 . 3 . x = 12x.

2x 3x 4

needs needs needs . Find missing factors.

2 . 3 2 . 2 3 . x

(x + 1)6 + (2x - 4)4 + 10 . 3x = . Multiply numerator and

2x . 6 3x . 4 4 . 3x denominator by missing factors.

6x + 6 + 8x - 16 + 30x = . Simplify.

12x 12x 12x

44x - 10 = . Combine numerators for

12x simpler form.

12(22x - 5) = . Factor and reduce, if

12 . 6x possible.

22x - 5 . Simplify the answer.

6x

94

Lesson 7.1

HOMEWORK

Add; then simplify if possible:

1. 4 + 7

a a

2. 2x + 7x

5 5

3. 7x + x

3 3

4. 5 + 3

12x 12x

5. 7a + 8a

18 18

6. a + -7

3a - 21 3a - 21

7. m + -3

m2 - 5m + 6 m2 - 5m + 6

8. 3r + -12

r2 - 4r r2 - 4r

9. 3a + 5

3a2 + 11a + 10 3a2 + 11a + 10

10. 2x + -3

4x2 + 4x - 15 4x2 + 4x - 15

95

Lesson 7.1

11. x + x

2 6

12. 3x + x

15 5

13. 2a + a

3 9

14. c + 4c

12 6

15. r + r

6 3

16. 2x + 3x

12 4

17. m + 3m + 1

8 2 4

18. 5m + 2m + 1

4 3 8

19. r + 1 + r + 5

2r 5r

20. 3b + 7 + b - 1

4b 8b

21. 3a + a + 5

5 10 4

22. n - 1 + 2n - 1

7 3

23. 2k - 3 + 3k + 1 + 4k - 2

6 3 2

24. 5r - 4 + 3r - 1 + 7r + 4

4r 8r 2r

25. 3 + 2x - 1 + 3x + 5

24x 6x 4x

96

Lesson 7.2 Adding Rational Expressions with Polynomial Denominators Unit 7

Rule: Add rational expressions with polynomial denominators by

the same methods used in the previous lesson:

1. Factor all denominators.

2. Select the LCM of all denominators for the LCD.

3. Compare each denominator to this LCM to determine the

missing factors from each.

4. Multiply both numerator and denominator of each

expression by its missing factor(s) to obtain the LCM as a

common denominator (LCD).

5. Combine numerators into a single rational expression

with the LCD as its denominator.

6. Simplify; reduce if possible.

Examples: 1. Add:

5 + 4 + 1 =

x2 + x - 6 x - 2 x + 3

5 + 4 + 1 = . Only x2 + x - 6 factors.

(x - 2)(x + 3) x - 2 x + 3 LCM is (x - 2)(x + 3).

LCM needs needs . Determine missing (x + 3) (x - 2) factors.

5 + 4(x + 3) + 1(x - 2) = . Multiply by missing

(x-2)(x+3) (x-2)(x+3) (x+3)(x-2) factors.

5 + 4x + 12 + x - 2 = . Expand numerators.

(x-2)(x+3) (x-2)(x+3) (x+3)(x-2)

5 + 4x + 12 + x - 2 = .Combine numerators.

(x - 2) (x + 3)

5x + 15 = . Simplify numerator.

(x - 2)(x + 3)

5(x + 3)1 = . Factor and reduce.

(x - 2)(x + 3)1

5 . Simplify the answer.

x - 2

97

Lesson 7.2

2. Add: -6x + 2x + 9 =

3x2 + 12x x + 4

-6x + 2x + 9 = . Factor; LCM is 3x(x + 4).

3x(x + 4) x + 4

LCM needs 3x . Determine missing factors.

-6x + 3x(2x + 9) = . Multiply by missing factors,

3x(x + 4) 3x(x + 4) 3x.

-6x + 6x2 + 27x = . Expand and combine

3x(x + 4) numerators.

6x2 + 21x = . Simplify numerator.

3x(x + 4)

13x(2x + 7) = . Factor and reduce.

13x (x + 4)

2x + 7 . Simplify the answer.

x + 4

3. Add: 3 + x - 4 =

4x2 - 9 2x2 - x - 3

3 + x - 4 = . Factor; LCM is

(2x+3)(2x-3) (2x-3)(x+1) (2x+3)(2x-3)(x+1).

needs (x+1) needs (2x+3) . Determine missing

factors.

3(x + 1) + (x - 4)(2x + 3) = . Multiply by missing

(2x+3)(2x-3)(x+1) (2x-3)(x+1)(2x+3) factors.

3x + 3 + 2x2 - 5x - 12 = . Expand numerators.

(2x+3)(2x-3)(x+1) (2x-3)(x+1)(2x+3)

3x + 3 + 2x2 - 5x - 12 = . Combine numerators.

(2x + 3)(2x - 3)(x + 1)

2x2 - 2x - 9 . Simplify numerator. 2x2 - 2x - 9 won't

(2x + 3)(2x - 3)(x + 1) factor; this is the simplified answer.

98

Lesson 7.2

HOMEWORK

Add; simplify, if possible: (Leave answers in factored form.)

1. a + 5

a + 2 a

2. 8 + 9

3x - 3 x2 - 2x + 1

3. m + 7

m2 - 49 x - 7

4. 4 + 8r + 15

r + 5 r2 + 5r

5. 2 + 1

a2 + 2a a + 2

6. 4 + 3

y - 4 4y - 16

7. 3 + 6

x + 3 x2 + 4x + 3

8. 4 + 2

n2 - 6n + 8 n - 2

9. 1 + 2

t2 + 5t + 6 t2 + 8t + 15

10. 2 + 3 + 5

c2 - c - 2 c - 2 c + 1

99

Lesson 7.2

11. 4 + -2

r2 - 4 r2 + 4r + 4

12. -16 + 4

a2 - 4a a - 4

13. m + 2 + 2m - 1

m - 8 m2 - 2m - 48

14. 5x + 2 + -2x - 5

4x - 8 x2 - x - 2

15. x - 4 + x

x2 - 2x - 15 x2 + 4x + 3

16. 5 + 3r - 3

r2 - 3r - 10 r2 + 7r + 10

17. m + 3m - 3

m2 - 9 m2 - m - 6

18. 2k + 4

k2 - 7k + 10 k2 - 25

19. 2d + d + 4

d2 - 9 d2 + 4d - 21 d2 + 7d

20. x2 - 6x - 2 + 5

2x2 + 3x + 1 x + 1

100

Lesson 7.3 Adding Other Types of Rationals Unit 7

Rule: When adding an integer or a polynomial to a rational

expression, write the integer or polynomial in rational form

(place it over 1) and then add as shown in previous lessons.

Examples: 1. Add: 5 + 2 =

3x

5 + 2 = . Write 5 in rational form, 5/1.

1 3x . LCD is 3x.

needs 3x LCD . Determine missing factors.

5 . 3x + 2 = . Multiply by 3x in numerator

1 . 3x 3x and denominator.

15x + 2 . Combine numerators. This

3x result cannot be simplified.

2. Add: 3 + m - 2 =

2m + 1

3 + m - 2 = . Place polynomial over 1.

2m + 1 1 . LCD is 2m + 1.

LCD needs . Determine missing factors.

(2m + 1)

3 + (m - 2)(2m + 1) = . Multiply to get the LCD for

2m + 1 1(2m + 1) both rational expressions.

3 + 2m2 - 3m - 2 = . Multiply in the numerator.

2m + 1 2m + 1

2m2 - 3m + 1 = . Add numerators.

2m + 1

(2m - 1)(m - 1) . Answer. The numerator factors,

2m + 1 but the expression won't reduce.

101

Lesson 7.3

Practice: A. Add: 2 + 2x - 1 B. Add: 3 + x + 5

3x + 2 2x - 3

Rule: If no denominator will factor, use the product of all

denominators as the LCD.

Example: 3. Add: 3 + 5 + 1 =

k + 2 k - 3 2k - 1

3 + 5 + 1 = . No denominator factors;

k + 2 k - 3 2k - 1 LCD is (k + 2)(k - 3)(2k - 1).

needs needs needs . Determine missing factors.

(k - 3) (k + 2) (k + 2)

&(2k - 1) &(2k - 1) &(k - 3)

3(k-3) (2k-1) + 5(k+2) (2k-1) + k(k+2) (k-3) = . Multiply by

(k+2)(k-3)(2k-1) (k-3)(k+2)(2k-1) (2k-1)(k+2)(k-3) missing factors.

6k2 - 21k + 9 + 10k2 + 15k - 10 + k2 - k - 6 = . Multiply in

(k+2)(k-3)(2k-1) (k-3)(k+2)(2k-1) (2k-1)(k+2)(k-3) numerators.

17k2 - 7k - 7 . Add numerators; this is

(k + 2)(k - 3)(2k - 1) the simplified answer.

Rule: Add rational expressions having monomial denominators

involving powers in the same manner as other rational

expressions: Every factor from the denominator of each

rational expression being added must appear as a factor of the

LCD.

102

Lesson 7.3

Example: 4. Add:

3 + 5 + 1 =

2a 3a2 6a3

3 + 5 + 1 = . Factor denominators;

2a 3a. a 2.3. a. a. a LCD is 2 . 3 . a . a . a = 6a3.

needs needs LCD . Determine missing factors.

3a2 2a

3 . 3a2 + 5 . 2a + 1 = . Multiply by missing factors.

2a . 3a2 3a2 . 2a 6a3

9a2 + 10a + 1 = . Multiply and combine

6a3 numerators.

(9a + 1)(a + 1) . Factor to see if reducing is

6a possible; this is the simplest

answer.

5. Add: 2 - m + 3m + 1 + m - 1 =

2m 15m3 5m2

2 - m + 3m + 1 + m - 1 = . Expand denominators if

2m 15m3 5m2 necessary; LCD is 30m3.

needs needs needs . Determine missing factors.

15m2 2 6m

(2-m)(15m2) + (3m+1)(2) + (m-1)(6m) = . Multiply by

(2m)(15m2) (15m3)(2) (5m2)(6m) missing factors.

30m2 -15m3 + 6m + 2 + 6m2 - 6m = . Multiply in

30m3 30m3 30m3 numerators and denominators.

-15m3 + 36m2 + 2 = . Combine numerators

30m3

_ 15m3 - 36m2 - 2 . Since the result won't

factor, this is 30m3 the simplest answer.

103

Lesson 7.3

HOMEWORK

Add. Simplify, if possible:

1. 3 + 2 14. 5 + 2 + 5 x x x2

2. 5 + 3 15. 3r + 5

8m 2r

3. 6 + 3 16. 3 + 7 + 9

r d2 d

4. 2 + p 17. 2 + 5 - 3m + 2 + 3m

p 5m 15m2 3m3

5. 3 + 4 18. 3 - k + 6 + 2k + 2 + 3m

x - 4 x - 6 2k 14k3 7k2

6. 7 + 2 19. t + 5 + 3

m + 3 m - 2 t - 2

7. 6 + 2 20. 3c - 1 + 3c

2a - 3 3a + 1 c + 5

8. 3 + 7 21. 5 + x + 3

r - 1 2r + 5 2x - 5

9. 3 + -5 22. a - 6 + a - 1

x - 9 x + 2 2a + 3

10. 5 + 11 23. f + 3 + 2f - 4

m - 3 m + 2 f + 5

11. 1 + 2 + 4 24. 2 + 5 + 7

g2 g g3 p - 1 p + 3 p - 2

12. 5 + 7 + 2 25. 3 + x2 - 2x + 4

2k3 3k 6k2 x - 5

13. 3 + -2 + 4

15a 5a2 3a2

104

( )

Lesson 7.4 Subtracting Rational Expressions Unit 7

Rule: To subtract a rational expression, add its opposite:

a - c means a + _ c

b d b d

Since _ c = -c, then a - c means a + -c .

d d b d b d

Examples: 1. Subtract:

5 - 3 =

x2 - x - 6 x + 2

5 + -3 = . Change to addition of the

x2 - x - 6 x + 2 opposite.

5 + -3 = . Factor denominator.

(x + 2)(x - 3) (x + 2) . LCD is (x - 3)(x + 2).

5 + -3 (x - 3) = . Multiply by (x - 3).

(x + 2)(x - 3) (x + 2)(x - 3)

5 + -3x + 9 = . Distribute - 3.

(x + 2)(x - 3) (x + 2)(x - 3)

5 - 3x + 9 = . Combine numerators

(x + 2)(x - 3) and simplify.

-3x + 14 . Answer.

(x + 2)(x - 3)

105

Lesson 7.4

2. Subtract:

3 - 3r - 1 =

2r

3 + -(3r - 1) = . Change to addition; 3 = 3/1.

1 2r

3 + -3r + 1 = . Distribute the negative.

1 2r

3 . 2r + -3r + 1 = . LCD is 2r.

1 . 2r 2r

6r + -3r + 1 = . Multiply by 2r.

2r 2r

6r - 3r + 1 = . Combine numerators

2r and simplify.

3r + 1 . Answer.

2r

Rule: Always reduce an answer, if possible, when adding or

subtracting rational expressions.

106

Lesson 7.4

Example: 3. Subtract:

c2 - 24 - c - 6 =

c2 - c - 6 c - 3

c2 - 24 + -(c - 6) = . Change to addition of the

c2 - c - 6 c - 3 opposite of the whole quantity.

c2 - 24 + - c + 6 = . - (c - 6) = -c + 6.

(c - 3)(c + 2) c - 3 . Factor denominator.

c2 - 24 + (-c + 6)(c + 2) = . LCD is (c - 3)(c + 2), so

(c - 3)(c + 2) (c - 3)(c + 2) multiply by (c + 2).

c2 - 24 + -c2 + 4c + 12 = . Multiply in numerator.

(c - 3)(c + 2) (c - 3 )(c + 2)

c2 - 24 - c2 + 4c + 12 = . Combine numerators.

(c - 3)(c + 2)

4c - 12 = . Simplify numerator.

(c - 3)(c + 2)

4(c - 3)1 = . Factor and reduce for

1(c - 3)(c + 2) simplified answer.

4 . Answer.

c + 2

107

Lesson 7.4

HOMEWORK

Subtract. Simplify, if possible:

1. 3 - 5 14. a2 - 17 - a - 5

x2 - 16 x + 4 a2 + 2a -3 a - 1

2. 2 - 5 15. r2 - 24 - r - 6

x2+5x-14 x - 2 r2 - r - 6 r - 3

3. 5 - 1 16. x2 + 9x + 8 - x + 3

m2-5m+6 m - 3 x2 + 2x - 8 x - 2

4. 5 - 3x - 2 17. c - 5 - c2 - 17

2x c - 1 c2 + 2c - 3

5. 11 - 2r - 1 18. m - 2 - m2 + 2

5r m + 1 m2 + 5m + 4

6. 3 - 3x - 2 19. k2 - 8 - k + 1

2x k2 - 8k + 12 k - 6

7. 3 - x - 3 20. 3 - 5x - 1

x - 5 x2 - 25 x + 3 x + 2

8. k - 2 21. a + 2 - a - 1

k2 - 4k + 3 k - 1 a + 1 a2 - 6a - 7

9. 2 - 4 22. r + 3 - r2 + 3r - 10

t2 - 2t - 35 t - 5 r - 8 r2 - 10r + 16

10. 3 - 2 23. 3c2 - 2c - 5

5y2 15y3 2c2 - 7c - 15 2c + 3

11. 3 - 4 24. x2 + 9x - 49 - x + 2

2m - 1 2m + 3 x2 - 7x x - 7

12. 3 - 5 25. d2 - 6

3x + 2 5x - 1 d4 - 17d2 + 16 d2 + 3d - 4

13. 3 - 6

8a3 12a2

108

Lesson 7.5 Simplifying by Factoring Out -1 Unit 7

Rule: To simplify an expression of the form a + c ,

x - b b - x

rewrite the denominator, b - x, in the practical form,

-1(x - b), by factoring -1 from it. Then put the -1 factor in the

numerator, -1 . c, (multiply numerator and denominator x - b by -1) and simplify as usual. The problem becomes:

a + - c

x - b x - b

Example: 1. Simplify:

-3x - 10 + -5 =

x2 - 5x 5 - x

-3x - 10 + -5 = . 5 - x = -1(-5 + x) = -1(x - 5).

x(x - 5) -1(x - 5)

-3x - 10 + 5 = . -5 = 5.

x(x - 5) x - 5 -1

-3x - 10 + 5. x = . LCD is x(x - 5).

x(x - 5) (x - 5)x . Multiply by x.

-3x - 10 + 5x = . Multiply and combine

x(x - 5) numerators.

2x - 10 = . Simplify numerator.

x(x - 5)

2(x - 5)1 = . Factor and reduce.

x(x - 5)1

2 . Simplified answer. x

Practice: A. Simplify: 9a + 20 + -a

a2 + 4a -4 - a

109

Lesson 7.5

Rule: Since subtracting an expression means the same as adding its

opposite, a - c may be written as either:

d

a + -c or a + c

d -d

Use whichever form is more convenient for simplifying the

expression.

Examples: 2. Simplify:

x2 - 33 - x + 7 =

x2 + 2x - 3 1 - x

x2 - 33 + (x + 7) = . Rewrite as a sum:

x2 + 2x - 3 - (1 - x) . a - c = a + c d -d

x2 - 33 + (x + 7)(x + 3) = . -(1 - x) = x - 1.

(x - 1)(x + 3) (x - 1)(x + 3) . LCD is (x - 1)(x + 3).

x2 - 33 + x2 + 10x + 21 = . Multiply; then combine

(x + 3)(x - 1) numerators.

2x2 + 10x - 12 = . Simplify numerator.

(x + 3)(x - 1)

2(x2 + 5x - 6) = . Factor 2 out.

(x + 3)(x - 1)

2(x + 6)(x - 1)1 = . Completely factor and

(x + 3)(x - 1)1 reduce.

2(x + 6) . Simplified answer.

x + 3

110

Lesson 7.5

Note: Two or more additions and/or subtractions may be

included in the same problem.

3. Simplify:

x2 + 5 - 1 =

x2 + 2x - 3 4x + 12 4x - 4

x2 + 5 + -1 = . Rewrite as a sum and

(x + 3)(x - 1) 4(x + 3) 4(x - 1) factor denominators.

4x2 + 5(x - 1) + -1(x + 3) = . Multiply by

4(x+3)(x-1) 4(x+3)(x-1) 4(x-1)(x+3) missing factors.

4x2 + 5x - 5 + -x - 3 = . Expand

4(x+3)(x-1) 4(x+3)(x-1) 4(x-1)(x+3) numerators.

4x2 + 4x - 8 = . Combine numerators

4(x + 3)(x - 1) and simplify.

14(x + 2)(x - 1)1 = . Completely factor.

14(x + 3)(x - 1)1

x + 2 . Simplified answer.

x + 3

111

Lesson 7.5

HOMEWORK

Simplify:

1. 3x + 2x 11. 3 + 2x

x - 7 7 - x x - 2 4 - 2x

2. 5 - 3m 12. 4c + d + 2d - 3c - d - c

2m - 8 4 - m 2cd 2cd 2cd

3. 6a - 3a 13. 4a - 3 - 1 - a

a2 - 49 7 - a 5 10

4. k2 + 2 - -3 14. m2 + 3m + 15 - 2 + 5

k2 - 5k + 4 1 - k m2 + 5m - 24 3 - m m + 8

5. r2 + 94 - 49 + r + 2 15. 3 - 2x - x

r2 - 7r 7 - r 3x - x2 3 - x

6. 4x + 2 16. 5 + p - 2

x2 - 25 5 - x p2 -5p+6 p2 -3p+2 p2 -4p +3

7. 10c + 5 17. 3x - 2 + 3

c2 - 9 3 - c x2 + 6x - 16 2 - x x + 8

8. 25 - 1 - -2 18. 13 + 2d - d + 2

t2 - 5 - 30 6 - t -28 + 11d - d2 7 - d

9. 2n - n 19. 1 - 3 + 2

n2 - 6n + 8 2 - n a + 6 36 - a2 a2 + 2a - 24

10. a + 5 - 5 20. 3 - -1 - 2

a - 5 5 - a 2 - a x2 - y2 3xy-x2-2y2 x2-xy-2y2

112

Lesson 7.6 Complex Rational Expressions Unit 7 Rules: A complex fraction is one which contains a fraction in the

numerator or denominator or both:

2 3 1 + 5

3 2 8 2 7 5 2 3 - 1

9 7 7 4

Likewise, a complex rational expression is one which contains a

rational expression in the numerator or denominator or both:

3a + 2 2a + 1 a

a 3 3 2 + 3 6 + a a 2a + 1 5 + 4a 7 5 2a 6

Complex rational expressions may be simplified by either of

two methods:

A. Follow the order of operations:

1. Simplify numerator completely.

2. Simplify denominator completely.

3. Divide numerator by denominator.

- or -

B. Multiply both numerator and denominator of the

expression by the LCD of all the rational expressions

which appear in the original complex rational expression.

Examples: 1. Simplify: 2

3

3

4

Method A

2 . Numerator and denominator are simple.

3 =

3

4

2 ÷ 3 = . Divide.

3 4

2 x 4 = 8 . Answer.

3 3 9

113

Lesson 7.6 Method B

2 . LCD for 2/3 and 3/4 is 12.

3 =

3

4

2 . 12 . Multiply numerator and denominator

3 = by LCD, 12.

3 . 12

4

24

3 = 8 . Answer.

36 9

4

2a + 1

2. Simplify: 3 a + 8

4 15

Method A

2a + 1 . 2a = 2a

1 3 = 1

a + 8

4 15

6a + 1 . LCD for 1 and 3 is 3.

3 3 =

15a + 32 . LCD for 4 and 15 is 60.

60 60

6a + 1 . Add numerator fractions.

3 =

15a + 32 . Add denominator fractions.

60

6a + 1 ÷ 15a + 32 = . Divide.

3 60

6a + 1 . 6020 = . Multiply by reciprocal.

31 15a + 32

120a + 20 . Answer.

15a + 32

114

( )

( )

( )

( ) ( )

(60)

(60)

(60) (60)

(60)

Lesson 7.6

Method B

2a + 1 . LCD for 1, 3, 4 and 15 is 60.

3 =

a + 8

4 15

2a + 1

3 = . Multiply by LCD, 60, in numerator

a + 8 and denominator.

4 15

2a(60) + 1

3 = . Distribute 60 and simplify.

a + 8

4 15

120a + 60

3 =

60a + 480

4 15

120a + 20 . Answer.

15a + 32

115

( ) ) (

( ) ) ( ( ( ) )

Lesson 7.6

3. Simplify:

1 - 4 - 45 . LCD is x2. Use Method B.

x x2 =

2 + 7 - 15

x x2

1 - 4 - 45 (x2) x x2 = . Multiply by LCD, x2, in

2 + 7 - 15 (x2) numerator and denominator.

x x2

1x2 - 4 x2 - 45 x2

x x2 = . Distribute x2.

2x2 + 7 x2 - 15 x2

x x2

x2 - 4x2 - 45x2

x x2 =

2x2 + 7x2 - 15x2

x x2

x2 - 4x - 45 = . Simplify.

2x2 + 7x - 15

(x - 9)(x + 5)1 = . Factor and reduce.

(2x - 3)(x + 5)1

x - 9 . Answer.

2x - 3

116

Lesson 7.6

HOMEWORK

Simplify:

1. 3 7. x - 9

4 x 9 x2 + 4x + 3

8 x

2. 4 - 2 8. c - 2

3 c - 1

3 + 2 c2 + 2c - 8

3 c - 1

3. x - x 9. y + 1 - 6

63 y x + x y - 1 - 12

3 y

4. 2a - 1 10. m + 3n

2 2n

2a + 1 2m - n

2 4n2

5. 1 - 1 11. 2 + 1

3x a

x - 1 2 - 1 - 1

9x a a2

6. 5 - 1

a

25 - 1

a2

117

Lesson 7.6

12. 1 + 2 18. 4 - r - 45

b r r2

a - a 2 + r - 15

b r r2

13. c - d 19. 1 + 7t + t2

4d c 2 12 6

c - 2d 1 + 1

8c2d2 2t t2

14. a2 - a 20. 1 - 13 + 36

b 2 a2 a4

2a2 + a 1 - 1 - 6

b 2 a2 1a3 a4

15. x - 6 + 5

x

1 - 1

x

16. m + m

3

4m

9

17. x2 - 4x + 4

x3

x - 2

2

118

Lesson 7.7 Complex Rational Expressions: Polynomial Denominators Unit 7

Rule: To simplify a complex rational expression with one or more

polynomial denominators:

1. Factor the denominator for each rational expression.

2. Select the LCD of all the denominators.

3. Multiply the numerator and denominator of the complex

rational expression by the LCD found in Step 2.

4. Simplify the resulting expression as usual.

119

( ) ( )

) )

( (

( (

) )

Lesson 7.7

Example: 1. Simplify:

m - 1

2m - 4 =

2 + 2

m - 2

m - 1 . Factor denominator.

2(m - 2) = . LCD is 2(m - 2).

2 + 2

m - 2

m -1 [2(m - 2)]

2(m - 2) = . Multiply numerator and

2 + 2 [2(m - 2)] denominator by 2(m - 2).

m - 2

m [2(m - 2)] - 1 [2(m - 2)] . Distribute 2(m - 2).

2(m - 2) =

2 [2(m - 2)] + 2[2(m - 2)]

m - 2

m [2(m - 2)]1 - 1 [2(m - 2)] . Cancel.

2(m - 2)1 =

2 [2(m - 2)] + 2[2(m - 2)]

m - 21 1

m - 2(m - 2) = . Simplify.

4 + 4(m - 2)

m - 2m + 4 = . Expand.

4 + 4m - 8

-m + 4 = . Simplify. This answer will do.

4m - 4

-(m - 4) = . Factor to try to reduce.

4(m - 1)

m - 4 . Better answer (negative

4(m - 1) fraction).

120

) )

) )

) )

( (

( (

( (

Lesson 7.7

2. Simplify: 6 + 5y + 2

y2 - 1 =

2 + 1

y - 1

6 + 5y + 2

(y + 1)(y - 1) = . Factor denominators.

2 + 1 . LCD is (y + 1)(y - 1).

y - 1

6 + 5y + 2 (y+1)(y-1) . Multiply by (y + 1)(y - 1)

(y+1)(y-1) = in numerator and

2 + 1 (y + 1)(y - 1) denominator.

y - 1

6(y+1)(y-1) + 5y + 2 (y+1)(y-1)

(y+1)(y-1) = . Distribute (y + 1)(y - 1).

2(y+1)(y-1) + 1 (y+1)(y-1)

y - 1

6(y+1)(y-1) + 5y + 2 (y+1)(y-1)1

(y+1)(y-1)1 = . Cancel.

2(y+1)(y-1) + 1 (y+1)(y-1)1

y - 11

6(y + 1)(y - 1) + 5y + 2 =

2(y + 1)(y - 1) + (y + 1)

6y2 - 6 + 5y + 2 = . Multiply.

2y2 - 2 + y + 1

6y2 + 5y - 4 = . Simplify.

2y2 + y - 1

(3y + 4)(2y - 1)1 = . Factor and reduce.

1(2y - 1)(y + 1)

3y + 4 . Answer.

y + 1

121

( (

( (

( (

( (

( (

) )

) ) )

) )

Lesson 7.7

3. Simplify:

a + 4 - 7

a a + 3 =

a + 3 + 3

a2 + 3a a + 3

a + 4 - 7

a a + 3 = . Factor denominators.

a + 3 + 3 . LCD is a(a + 3).

a(a + 3) a + 3

a+4 7 a(a+3) a a+3 = . Multiply by LCD in (a+3) + 3 a(a+3) numerator and

a(a+3) a+3 denominator.

a+4 a(a+3) 7 a(a+3) . Multiply all rational

a a+3 = expressions by a(a + 3).

(a+3) a(a+3) + 3 a(a+3)

a(a+3) a+3

1 1

a+4 a(a+3) 7 a(a+3) . Cancel.

a 1 a+31 =

(a+3) a(a+3)1 + 3 a(a+3)1

a(a+3)1 a+31

(a + 4) (a + 3) - 7a = . Simplify.

a + 3 + 3a

a2 + 7a + 12 - 7a = . (a + 4)(a + 3) = a2 + 7a + 12.

4a + 3

a2 + 12 . Answer won't reduce.

4a + 3

122

Lesson 7.7

HOMEWORK

Simplify:

1. a 6. r + 2r - 1

a - b r - 1

1 + b r + 1

a - b r - 1

2. 2x - 1 7. x - 1

x - y x + 1

x - 1 x - 2x - 1

x - y x + 1

3. a - 2b2 8. r - 2

a + b 6r - 3

a + ab - 3b2 r2 - 4

a + b 2r2+3r-2

4. c - c - d 9. 3 - 12

1 + cd a a2 + 2a

1 + c2 - cd 2 - 1

1 + cd a + 2

5. 6 + 5x + 2 10. 7x

x2 + 1 2x2 - 5x - 3

2 + 1 6 + 2

x - 1 x - 3 2x + 1

123

Lesson 7.7

11. x + 3 16. 2y + 4 - y - 1

x - 3 y + 8 y - 2

3x + 9 y2 - 49

x2 – 9 y2 + 6y - 16

12. ac – ad 17. g - 3 - 1

c2 - d2 g + 3

c - a g + 3 + 2

c + a g - 3 g - 3

13. m + 2 - 12 18. r - 1 - r

m + 3 1 + r r

m - 5 + 16 r + 1 - r

m + 3 1 + r r

14. x2 + y2 19. x2 - 49

x2 - y2 x2 + 6x - 16

x - y - x + y 2x + 4 - x - 3 x + y x – y x + 8 x - 2

15. a + 2a + 1 20. 1 + 1

a - 1 x + 5 x - 3

a + 2 2x2 - 3x - 5

a – 1 x2 + 2x - 15