unit: polynomials: multiplying and...
TRANSCRIPT
Page 1
Intro to Applied and PreCalc – Notes Unit: Polynomials and Factoring
Name ________________________
Unit: Polynomials:
Multiplying and Factoring
Specific
Outcome
10I.A.1 Demonstrate an understanding of factors of whole
numbers by determining:
Prime factors
Greatest common factor
Least common multiple
10I.A.3 Demonstrate an understanding of powers with
integral and rational exponents
10I.A.4 Demonstrate an understanding of the multiplication
of polynomial expressions
10I.A.5 Demonstrate an understanding of common factors
and trinomial factoring
Comments : ________________________________________________
_____________________________________________________________
_____________________________________________________________
_____________________________________________________________
Page 2
Intro to Applied and PreCalc – Notes Unit: Polynomials and Factoring
Review Lesson on Polynomials
Naming Polynomials Polynomials – by number of terms
Monomials (1 term)
Binomials (2 terms)
Trinomials (3 terms)
Polynomial with 4 terms
Polynomial with 5 terms
Etc.
Polynomials – by highest degree
Constant (no variable visible)
Linear (variable with exponent of 1)
Quadratic (variable with exponent of 2)
Cubic (variable with exponent of 3)
Quartic (variable with exponent of 4)
Quintic (variable with exponent of 5)
Sixth degree polynomial (variable with exponent of 6)
Etc.
Collecting and Combining “Like” Terms When collecting like terms, they must have the same variable and same exponent.
Add or subtract the coefficients (NOTE: the + and – signs belong to the term behind them).
Examples: Simplify the following
a) 6xy + 4x + 8xy b) 2x – 3y – 6x + 2y c) 3x + 5x2 – 7y – 8x + 8y
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Intro to Applied and PreCalc – Notes Unit: Polynomials and Factoring
Adding and Subtracting Polynomials
Adding
1) Remove the brackets
2) Simplify by collecting like terms
3) Combine like terms
4) Arrange in alphabetical order and then from highest to lowest power
Example 1: Simplify (x2 + 4x + 3) + (2x2 + 5x + 1)
Subtracting
1) Remove the first set of brackets (unless there is a number in front of it other than +1).
NOTE: a subtraction sign is similar to -1.
2) Multiply the negative one through the second bracket (take the opposite for each term in
the second set of brackets) and remove the brackets.
3) Collect like terms
4) Combine like terms
Example 2: Simplify (4x2 – 5x + 7) – (3x2 + 2x – 4)
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Intro to Applied and PreCalc – Notes Unit: Polynomials and Factoring
Multiplying and Dividing Monomials
Multiplying (coefficient coefficient)(variable variable)
remember to add exponents when multiplying powers
Examples: Find the product
a) (30x)(4y) b) -5x2(-3yz)
c) 4(3x – 2y + 7)
use the distributive property to expand each expression multiply each term inside the
brackets by the term outside the brackets
Dividing divide each term of the polynomial by the monomial (note example 3)
remember to subtract exponents for common variables
Examples: Simplify a) 12x
3
b) -18a3b5c7
6ab2c4
c) 8x5- 16x4 + 4x2 – 2x3
2x2
Assignment: Adding and Subtracting Polynomials handout
Multiplying and Dividing Monomials handout
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Intro to Applied and PreCalc – Notes Unit: Polynomials and Factoring
Lesson 1: Multiplying Polynomials (Part 1)
Multiplying Monomial by Polynomial: Distribution Method
Examples: Multiply the following polynomials
a) 5y2(x2 - y) b) 4y(2y2 + 3y - 1)
Multiplying Binomial by Binomial: FOIL Method
A technique for multiplying two binomials is using the F.O.I.L. method.
The letters F. O. I. L. stand for , , ,
We always multiply these terms.
Steps :
1) Identify the first term in each bracket and them together.
2) Identify the most outside terms of the expression and multiply them together.
3) Identify the most inside terms of the expression and multiply them together.
4) Identify the last term in each bracket and multiply them together.
5) Simplify by collecting like terms.
Examples: Multiply the following polynomials
c) (x +2)(x + 5)
First Outer Inner Last
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Intro to Applied and PreCalc – Notes Unit: Polynomials and Factoring
Multiply the following polynomials
d) (x + 6)(x + 8) e) (2x - y)(3x + y)
f) (x - 2y)(x + 2y)
Binomial Squared:
g) (x + 5)2 *This is the same as (x + 5)(x + 5)
h) (2x - y)2
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Intro to Applied and PreCalc – Notes Unit: Polynomials and Factoring
Assignment #1 FOIL
Multiply the following binomials using FOIL:
1) 3 4x x 2) 2 5x x
3) 2 1 3 2x x 4) 4 3 2x x
5) 4x 2
6) 3 3 4x x
7) 4 2 4x x 8) 5 4t t
9) 3 2 2 9w w 10) 2 2z z
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Intro to Applied and PreCalc – Notes Unit: Polynomials and Factoring
11) a b 2 12) 5 5 6 1e e
13) 2 1x x 14) 5x 2
15) 7 3x y x y 16) 6 2x y x y
17) 4x y 2 18) 8 7x x
19) 3 4x y x y 20) 1 10x x
21) 8x y 2 22) 8x 2
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Intro to Applied and PreCalc – Notes Unit: Polynomials and Factoring
Lesson 2: Multiplying Polynomials (Part 2)
Multiplying Binomial by Trinomial: Distribution Method
Examples: Multiply the following polynomials
a) (y - 3)(y2 - 4y + 7)
b) (2x - 1)(2x2 + 5x - 3)
Examples: Expand or simplify the following polynomials
a) 3(x - 1)(2x - 3)
b) (5a + 4) + (a – 1)(a + 2) – (2a – 3)
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Intro to Applied and PreCalc – Notes Unit: Polynomials and Factoring
Assignment #2 Distribution Method
Multiply the following polynomials
1) 23 2 5 2d d d 2) 24 5 9 1s s s
3) 23 6 4 7c c c 4) 25 7k k k
5) 2 25 2 2 6y y y y 6) 2 25 3 3 4 5r r r r
7) 2 22 4 2 7c c c c
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Intro to Applied and PreCalc – Notes Unit: Polynomials and Factoring
Simplify the following polynomials:
1) 4 5 3 2 3 3 1y y y 2) 3 9 2 5 4 7 6 3a a a a
3) 2 3 5 6 5 4d e d e d e d e 4) 2
5 4 2 7 8 6n n n
5) 2 23 4 2 5 6w w w w 6) 2 4 5 2 3 5t s t s t s
7) 3 7 4 3 2 2a a a 8) 2
2 3 6 3b b b
9) 2 4 3x y x y x y x y 10) 2
4 6 2 3 2a c a c a c
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Intro to Applied and PreCalc – Notes Unit: Polynomials and Factoring
Lesson 3: Greatest Common Factor and Prime Factors
The greatest common factor (GCF) is the largest factor shared by two or more terms. This
is the largest number that divides evenly into two or more numbers.
In order to determine the GCF of 2 or more numbers, you must begin by listing all the factors
of those numbers.
When a factor of a number has exactly two divisors, one and itself, the factor is a prime
factor.
For example,
The factors of 12 are 1, 2, 3, 4, 6, and 12.
The prime factors of 12 are 1, 2, and 3.
To determine the prime factorization of 12, write 12 as a product of its prime factors: 2 x 2
x 3, or 2² x 3
NOTE:
o The first 10 prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29
o Natural numbers greater than one that are not prime, are composite.
Example 1: Write the prime factorization of 3300. Begin by using a Factor Tree.
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Intro to Applied and PreCalc – Notes Unit: Polynomials and Factoring
Example 2: List the factors of each of the following numbers. Then, identify the GCF.
a) 15 and 30
b) -24 and -48
Example 3: Determine the greatest common factor of 4xy and 2x²y.
Example 4: Determine the greatest common factor of the following sets of terms:
18x²yz, 27x²y²z, 9x²y²
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Intro to Applied and PreCalc – Notes Unit: Polynomials and Factoring
Least/Lowest Common Multiple
The least/lowest common multiple (LCM) is the smallest multiple shared by two or more
terms. To generate multiples of a number, multiply the number by the natural numbers; that
is, 1, 2, 3, 4, 5, and so on.
For example, some multiples of 26 are:
26 · 1 = 26 26 · 2 = 52 26 · 3 = 78
For two or more natural numbers, we can determine their least common multiple.
Example 5: Determine the least common multiple of 18, 20, and 30.
Example 6: Mei is stacking toy blocks that are 12 cm tall next to blocks that are 18 cm tall. What is
the shortest height at which the two stacks will be the same height?
Page 15
Intro to Applied and PreCalc – Notes Unit: Polynomials and Factoring
Assignment#3 Prime Factors, LCM’s and GCF’s
1. Two ropes are 48 m and 32 m long. Each rope is to be cut into equal pieces and all pieces
must have the same length that is a whole number of meters. What is the greatest possible
length of each piece?
a) List the factors of both numbers.
b) State the Greatest Common Factor (GCF).
2. Complete each factor tree and write the prime factorization for each number.
a) 144 b) 600
c) 252 d) 900
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Intro to Applied and PreCalc – Notes Unit: Polynomials and Factoring
3. Find the GCF for the following:
a. 44 and 70
b. 36 and 48
4. Find the Lowest Common Multiple (LCM) of the following:
a. 12 and 30
b. 16 and 18
5. Hamburger patties come in packages of 8. Buns come in packages of 6. What is the least
number of hamburgers that can be made with no patties or buns leftover?
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Intro to Applied and PreCalc – Notes Unit: Polynomials and Factoring
6. List the prime factors of the coefficients and the variable for each term and find the
Greatest Common Factor between the two terms.
a. 6x2, 12x
b. 20c2d3, 30cd2
c. 4b2c3 , 6bc2
d. 18xy2z, 24x2y3z2
e. 5m3n, 20mn2
7. Determine the GCF of the following sets of terms:
a. 14a, 21b b. -5n2, -10n
c. 3rs, 7t d. 12f2g3, 16fg2, 32f3g2
e. -15d2e3, -30cd2e, -45cde f. -18j3k, 27j2kl, 36j2k2l2
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Intro to Applied and PreCalc – Notes Unit: Polynomials and Factoring
Lesson 4: Common Factoring
Factoring is the process of . The better you are at
multiplying, the better you will be at factoring.
Multiplication Factoring
5x(x – 2y) = 5x2 – 10xy 5x2 – 10xy = 5x(x – 2y)
(x – 3)(x + 5) = x2 + 2x – 15 x2 + 2x – 15 = (x – 3)(x + 5)
1) Common Factoring: When factoring, begin by looking for the GCF. It could contain a number, a
variable or both.
Place this greatest common factor in front of parentheses, with the remaining polynomial
the parentheses.
Once this is done, the same number of terms as in the original question should be inside. (i.e. a
leaves a .)
Examples: Factor the following:
i) 4x + 8 = ( ) ii) 8xy – 32y2 = GCF Remaining factor
iii) 7n2 – 49n = iv) 15w3 + 5w =
iv) b – b2r3c = vi) 12n3 – 16n2 + 32n =
vii) 3x3 – 6x2y + 9xy2 =
Factoring can always be quickly and easily checked by the polynomials
together to see if the product is the original polynomial.
Page 19
Intro to Applied and PreCalc – Notes Unit: Polynomials and Factoring
Assignment#4 Factoring Binomials Factor the following polynomials.
a. 6s +30 b. 4t +28
c. 5a – 5 d. 216 12r r
e. 7 14 49xy xy xz f. 3 2 23 9 27c c d
f. 215 5w w h.
2 34 6a a
i. 2 210 50x y xy j.
22 4g g
k. 2 2 235 15 5x y x y xy l. 2 3 22 6 4r r s rs
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Intro to Applied and PreCalc – Notes Unit: Polynomials and Factoring
Lesson 5: Factoring ax2 + bx + c (leading coefficient)
Use this method anytime there is a in front of your x2 which
cannot be factored out.
Product-Sum-Factor (PSF) Method
Factor: 3x2 + 17x + 10
ALWAYS begin factoring by checking for common
.
From the remaining trinomial, calculate the product of
the and coefficients.
List all of the of this product.
(List them as pairs)
From these pairs of factors, identify which pair, when
added together, result in the middle term of the trinomial.
Re-write the trinomial with the original first and last
terms. In between these two terms, insert two new
terms using the pair of coefficients from step .
Now the first two and the
last two terms. (Notice how a common factor emerges.)
Now again. Collect the common
brackets.
You can your answer by expanding
(multiplying using FOIL)
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Intro to Applied and PreCalc – Notes Unit: Polynomials and Factoring
Examples: Factor the following fully, if possible:
1. 3y2 – 10y + 8 =
2. 8a2 + 18a – 5 =
3. 15x2 + 48x + 36=
4. 2c2 + 2c – 3 =
5. 5x2 – 20xy + 20y2 =
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Intro to Applied and PreCalc – Notes Unit: Polynomials and Factoring
6. 6b4 + 7b2 – 10 =
7. 2d3 + 7d2 – 30d =
8. 10g2 – 3gh – h2 =
9. 6k2 + 14km – 12m2 =
Page 23
Intro to Applied and PreCalc – Notes Unit: Polynomials and Factoring
Assignment #5 Factoring with leading coefficients. Factor the following trinomials:
1. 22 3 1x x 2. 23 5 2x x
3. 26 13 6x x 4. 22 5 12x x
5. 24 18 10x x 6. 23 17 28x x
7. 22 11 14r r 8. 22 11 12l l
9. 23 9 6w w 10. 210 8 2b b
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Intro to Applied and PreCalc – Notes Unit: Polynomials and Factoring
11. 2 25 6y yz z 12. 212 19 4a a
13. 22 7 15f f 14. 2 110r r
15. 26 6 3b b 16. 2 210 17 3m mn n
17. 2 29 9 2g gf f 18. 26 32 42l l
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Intro to Applied and PreCalc – Notes Unit: Polynomials and Factoring
Lesson 6: Trinomial Factoring (leading coefficient of 1)
Trinomials will factor to 2 brackets.
Example: x2 + 5x + 6
Steps: ALWAYS factor out any terms/variables first.
Identify the of the last term of the trinomial.
Next, determine which of factors either up to or to get
the middle term
Therefore, x2 + 5x + 6 factors to ( )( )
Examples: Factor the following trinomials fully, if possible:
1. x2 + 9x + 18 2. y2 – 2y - 15
3. 5 + b2 – 6b 4. a2 – 4a - 60
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Intro to Applied and PreCalc – Notes Unit: Polynomials and Factoring
5. x2 + 8xy + 16y2 6. p4 –2p2 – 15
7. 2x2 + 8x + 6 8. x2 - 7x - 6
9. 3x3 - 18x2 + 27x 10. 2x2yz3 - 10xyz3 - 48z3y
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Intro to Applied and PreCalc – Notes Unit: Polynomials and Factoring
Assignment #6 Factoring Trinomials
Factor the following trinomials:
1. 2 8 12y y 2. 2 10 21x x
3. 2 19 90a a 4. 2 242m mn n
5. 2 19 34b b 6. 2 10 24g g
7. 2 15 26n n 8. 2 15 56c c
9. 2 27 10s st t 10. 2 6 12f f
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Intro to Applied and PreCalc – Notes Unit: Polynomials and Factoring
Lesson 7: Factoring Difference of Squares
A) Perfect Square Binomials: ax2 - by2
A difference of squares has 3 main features:
1. The first term is a perfect .
2. The second term is a square.
3. They are separated by a sign.
The term is absent because it is .
Eg. x2 - 0xy - 16y2
Factoring a perfect square binomial results in two similar binomials,
that differ only in the sign.
To factor a difference of squares:
Remember to ALWAYS begin factoring by looking for a factor.
The first term of the binomials’ comes from the square of the
term.
The term of the binomials’ comes from the square of the second
term.
Place a sign in one parentheses and a in the other.
Check the result by using F.O.I.L .
Example from above: x2 - 16y2 = ( )( )
Examples:
1. x2 - 9
2. 225b2 - a2
3. 49 + x2
4. -y2 + 36
5. 3x3 - 48x
6. x4 - 16
Example:
x2 – 16y
2
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Intro to Applied and PreCalc – Notes Unit: Polynomials and Factoring
B) Perfect Square Trinomials: x2 bxy + cy2
A perfect square trinomial has main features:
1. The first term is a .
2. The term is a perfect square. The sign of the last term
is always .
3. The term can be either positive or negative. It is always double the
square root of the last term.
Factoring a perfect square trinomial results in two
Example: Factor: x2 - 8xy + 16y2
Check:
Examples: Factor the following trinomials fully, if possible.
1. 49 + 14x +x2
2. 5b3 - 40b2 + 80b
3. The volume of a rectangular prism is represented by 2x3 - 24x2 + 72x. What are possible
dimensions of the prism? (V = LxWxH)
Example:
x2 – 8xy + 16y
2
Page 30
Intro to Applied and PreCalc – Notes Unit: Polynomials and Factoring
Assignment #7 Factoring Difference of Squares and Perfect
Squares
Factor the following DOS’s and POS’s:
1. 2 5 25n n 2. 2 100a
3. 2 49t 4. 2 4x
5. 264 h 6. 2 29 16c d
7. 2 2r s 8. 2 250 72g h
9. 2 29 15p r 10. 2 144s
11. 2 272 32g h 12. 2 12 36y y
13. 2 6 9x x 14. 22 12 18z z
15. 2 5 25a a 16. 2144 48 4b b
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Intro to Applied and PreCalc – Notes Unit: Polynomials and Factoring
17. 2 216 64d e 18. 227 48m
19. 22 24 72k k 20. 3 23 51 147c c c
21. 2 2100 25a b 22. 3 218 81s t s t st
23. 481 16d 24. 2 212 12 3lnlm lmn
Page 32
Intro to Applied and PreCalc – Notes Unit: Polynomials and Factoring
ANSWER KEY:
Assignment #1
1. 2. 3. 4.
5. 6. 7. 8. 20
9. 10. 11. 12.
13. 14. 15. 16.
17. 18. 19. 20.
21. 22.
Assignment #2
1. 2. 3.
4. 5.
6. 7.
1. 2.
3. 4.
5. 6.
7. 8.
9. 10.
Assignment #3
1. GCF = 16
2. a) b) c) d)
3. a) GCF = 2 b) GCF = 12 4. a) LCM = 60 b) LCM = 144
5. LCM = 24
6. a) GCF = 6x b) GCF = 10cd2 c) GCF = 2bc2 d) GCF = 6xy2z e) GCF = 5mn
7. a) GCF = 7 b) GCF = -5n c) GCF = 1 d) GCF = 4fg2 e) GCF = -15de
f) GCF = 9j2k
Page 33
Intro to Applied and PreCalc – Notes Unit: Polynomials and Factoring
Assignment #4
a) 6(s + 5) b) 4(t + 7) c) 5(a – 1) d) 4r(4r – 3)
e) 7x(y + 2y – 7z) f) g) 5w(3w -1) h)
i) 10xy(xy – 5) j) 2g(g + 2) k) 5xy(7x + 3xy + 1) l) )
Assignment #5
1. (2x + 1)(x + 1) 2. (3x – 1)(x + 2) 3. (3x – 2)(2x – 3)
4. (2x – 3)(x + 4) 5. 2 (x – 5)(x + 1) 6. (3x – 4)(x + 7)
7. (2r + 7)(r + 2) 8. (2l + 3)(l + 4) 9. 3(w + 2)(w + 1)
10. 2(5b2 + 4b + 1) 11. (y + 3z)(y + 2z) 12. (4a + 1)(3a + 4)
13. (2f – 3)(f + 5) 14. (r + 11)(r – 10) 15. 3(2b2 + 2b – 1)
16. (2m – 3n)(5m – n) 17. (3g – 2f)(3g – f) 18. 2(3l + 7)(l + 3)
Assignment #6
1. (y + 6)(y + 2) 2. (x + 7)(x + 3) 3. (a – 10)(a – 9)
4. (m – 7n)(m + 6n) 5. (b + 2)(b + 17) 6. (g – 6)(g – 4)
7. (n – 13)(n – 2) 8. (c – 8)(c – 7) 9. (s – 5t)(s – 2t)
10. Not Factorable
Assignment #7
1. not factorable 2. (a – 10)(a + 10) 3. (t – 7)(t + 7)
4. Not factorable 5. (8 – h)(8 + h) 6. (3c – 4d)(3c + 4d)
7. (r – s)(r + s) 8. 2(5g – 6h)(5g + 6h) 9. 3(3p2 – 5r2)
10. Not Factorable 11. 8(3g – 2h)(3g + 2h) 12. (y + 6)2
13. (x – 3)2 14. 2(z + 3)2 15. Not Factorable
16. 4(6 – b)2