lesson 6.2 polynomial operations iansw.pdf · example 6.2.6 3z3 15z3 2zy2 5zy2 example 6.2.5 5x3...

CONCEPT 1:ADDING AND SUBTRACTINGPOLYNOMIALS

DefinitionsA monomial is an algebraic expression that contains exactly one term. The term may be a constant, or the product of a constant and one or morevariables. The exponent of any variable must be a nonnegative integer (thatis, a whole number).

The following are monomials:

12 x2 �5wy3 �12

�gt2 4.35T

A monomial in one variable, x, can be written in the form axr, where a isany real number and r is a nonnegative integer.

LESSON 6.2 POLYNOMIAL OPERATIONS I EXPLAIN 383

Concept 1 has sections on

• Definitions

• Degree of a Polynomial

• Writing Terms inDescending Order

• Evaluating a Polynomial

• Adding Polynomials

• Subtracting Polynomials

LESSON 6.2 POLYNOMIAL OPERATIONS I

Overview

In business, people use algebra everyday to find unknown quantities.

For example, a manufacturer may use algebra to determine a product’sselling price in order to maximize the company’s profit. A landscapearchitect may be interested in finding a formula for the area of a patio deck.

To find these quantities, you need to be able to add, subtract, multiply, and divide polynomials.

Explain

The following are not monomials:

�x23� The denominator contains a variable with a positive exponent.

So the term cannot be written in the form axr where r is a nonnegative integer.

�3x�2� There is a squared variable under a cube root symbol.

So the term cannot be written in the form axr where r is a nonnegative integer.

A polynomial is the sum of one or more monomials. Here are some examples:

2x3 � 5x � 2 x � 2 5 3xy2 � 7x � 5y � 1

A polynomial with one, two, or three terms has a special name.

NumberName of terms Examples

monomial 1 x, 5y, 3xy3, 5

binomial 2 x � 1, 2x2 � 3, 5xy3 � 4x3y2

trinomial 3 x2 � 2x � 1, 3x2y3 � xy � 5

Determine if each expression is a polynomial.

a. �4w3 b. � 3x c. �x�3� � 2 d. x � 2

Solution

a. The expression is a polynomial. It has one term, so it is a monomial. The term has the form awr, where a � �4 and r � 3.

b. The expression is not a polynomial.

The term �2x42� cannot be written in the form axr where r is a

nonnegative integer.

c. The expression is not a polynomial.The term �x�3� cannot be written in the form axr where r is a nonnegative integer.

d. The expression is a polynomial. It has two terms, so it is a binomial. Each term can be written in the form axr:

x � 2

1x1 � 2x0

24�x2

Example 6.2.1

384 TOPIC 6 EXPONENTS AND POLYNOMIALS

A polynomial with 4 terms is called a “four term polynomial.”A polynomial with 5 terms is called a “five term polynomial,” and so on.

Remember: x0 � 1, for x � 0x1 � x

Degree of a PolynomialThe degree of a term of a polynomial is the sum of the exponents of thevariables in that term.

For example, consider this trinomial: 6x3y2 � xy2 � 35x4.

The degree of the first term is 5. 6x3y2

degree � 3 � 2 � 5

The degree of the second term is 3. xy2 � x1y2

degree � 1 � 2 � 3

The degree of the last term is 4. 35x4

In 35, the exponent does not contribute degree � 4to the degree because the base, 3, is not a variable.

The degree of a polynomial is equal to the degree of the term with the highest degree.

In this polynomial, the term with degree 5 degree 3 degree 4the highest degree has degree 5. So this polynomial has degree 5. 6x3y2 � x1y2 � 35x4

The polynomial has degree 5.

Writing Terms in Descending OrderThe terms of a polynomial in one variable are usually arranged by degree,in descending order, when read from left to right.

For example, this polynomial x3 � 7x2 � 4x � 2contains one variable, x. The terms of the polynomial x3 � 7x2 � 4x1 � 2x0

are arranged by degree in descending order. degree degree degree degree

3 2 1 0


Arrange the terms of this polynomial in descending order and determinethe degree of the polynomial: 7x3 � 8 � 2x � x4

Solution

Write �8 as �8x0. Write 2x as 2x1. � 7x3 � 8x0 � 2x1 � x4

Arrange the terms by degree in � � x4 � 7x3 � 2x1 � 8x0

descending order (4, 3, 1, 0).

The last two terms may be written � � x4 � 7x3 � 2x � 8without exponents.

The term of highest degree is �x4. The degree of �x4 is 4. So, the degreeof this polynomial is 4.

Evaluating a PolynomialTo evaluate a polynomial, we replace each variable with the givennumber, then simplify.

Evaluate this polynomial when w � �3 and y � 2: 6w2 � 4wy � y4 � 5

Solution

Substitute �3 for w and 2 for y. � 6(�3)2 � 4(�3)(2) � (2)4 � 5

First, do the calculations with � 6(9) � 4(�3)(2) � 16 � 5the exponents.

Multiply. � 54 � 24 � 16 � 5

Add and subtract. � 19

Adding PolynomialsTo add polynomials, combine like terms. Recall that like terms are terms that have the same variables raised to thesame power. That is, like terms have the same variables with the sameexponents.

Like terms

3x, �12x

�8xy2, 5.6xy2

24, �11

4xy, 6yx

NOT like terms

7x, �5xy The variables do not match.

3x2y3, �2x3y2 The powers of x do not match.The powers of y do not match.

Example 6.2.3

Example 6.2.2


Find: (5x3 � 13x2 � 7) � (16x3 � 8x2 � x � 15)

Solution � (5x3 � 13x2 � 7) � (16x3 � 8x2 � x � 15)

Remove the parentheses. � 5x3 � 13x2 � 7 � 16x3 � 8x2 � x � 15

Write like terms next to � � xeach other.

Combine like terms. � 21x3 � 5x2 � x � 8

Find the sum of (3z3 � 2zy2 � 6y3) and (15z3 � 5zy2 � 4z2).

Solution

Write the sum. �(3z3 � 2zy2 � 6y3) � (15z3 � 5zy2 � 4z2)

Remove the parentheses. � 3z3 � 2zy2 � 6y3 � 15z3 � 5zy2 � 4z2

Write like terms next � � 6y3 � 4z2

to each other.

Combine like terms. � 18z3 � 3zy2 � 6y3 � 4z2

Subtracting PolynomialsTo subtract one polynomial from another, add the first polynomial to theopposite of the polynomial being subtracted.

To find the opposite of a polynomial, multiply each term by �1.

For example:

The opposite of 5x2 is �5x2.The opposite of �2x � 7 is 2x � 7.

Find: (�18w2 � w � 32) � (40 � 13w2)

Solution � (�18w2 � w � 32) � (40 � 13w2)

Change the subtraction to � (�18w2 � w � 32) � (�1)(40 � 13w2)addition of the opposite.

Remove the parentheses. � �18w2 � w � 32 � 40 � 13w2

Write like terms next to � � weach other.

Combine like terms. � �5w2 � w � 72

So, (�18w2 � w � 32) � (40 � 13w2) � �5w2 � w � 72.

� 32 � 40�18w2 � 13w2

Example 6.2.6

� 2zy2 � 5zy23z3 � 15z3

Example 6.2.5

� 7 � 15� 13x2 � 8x25x3 � 16x3

Example 6.2.4


We can also place one polynomial beneaththe other and add like terms.

5x3 � 13x2 � 7� 16x3 � 8x2 � x � 15

21x3 � 5x2 � x � 8

We can also place one polynomial beneaththe other and add like terms.

3z3 � 2zy2 � 6y3

� 15z3 � 5zy2 � 4z2

18z3 � 3zy2 � 6y3 � 4z2

Here’s a way to find the opposite of apolynomial:

Change the sign of each term.

We can also place one polynomial beneaththe other and subtract like terms.

�18w2 � w � 32� (�13w2 � 40)

To do the subtraction, we change the signof each term being subtracted, then add.

�18w2 � w � 32� (�13w2 � w � 40)

�5w2 � w � 72

Subtract (15z2 � 5yz2 � 4y3) from (6y3 � 10z3 � 2yz2).

Solution

Be careful! “Subtract A from B” means B � A. The order is important.

Write the difference. � (6y3 � 10z3 � 2yz2) � (15z2 � 5yz2 � 4y3)

Change the subtraction to addition of the opposite.

� (6y3 � 10z3 � 2yz2) � (�1)(15z2 � 5yz2 � 4y3)

Remove the parentheses.

� 6y3 � 10z3 � 2yz2 � 15z2 � 5yz2 � 4y3

Write like terms next to each other.

� 6y3 � 4y3 � 10z3 � 2yz2 � 5yz2 � 15z2

Combine like terms. � 2y3 � 10z3 � 7yz2 � 15z2

Here is a summary of this concept from Interactive Mathematics.

Example 6.2.7


We can also place one polynomial beneaththe other and subtract like terms.

6y3 � 10z3 � 2yz2

� (4y3 � 5yz2 � 15z2)

To do the subtraction, we change the signof each term being subtracted, then add.

6y3 � 10z3 � 02yz2

� (�4y3 � 05yz2 � 15z2)2y3 � 10z3 � 10yz2 � 15z2

CONCEPT 2:MULTIPLYING AND DIVIDINGPOLYNOMIALS

Multiplying a Monomial By a MonomialTo find the product of two monomials, multiply the coefficients. Then, usethe Multiplication Property of Exponents to combine variable factors thathave the same base.

Find: 7m3n4 � 6mn2

Solution

Write the coefficients next to each other. � 7m3n4 � 6mn2

Write the factors with base m next to each other, and write the factors � (7 � 6)(m3 � m1)(n4 � n2)with base n next to each other.

Use the Multiplication Property of � (7 � 6)(m3 � 1n4 � 2)Exponents.

Simplify. � 42m4n6

Find: �13

�w3x7y � 6w2y5

Solution

Write the coefficients next to each other. � �13

�w3x7y � 6w2y5

Write the factors with base w next to each other, and write the factors � ��

13

� � 6�(w3 � w2)(x7)(y1 � y5)with base y next to each other.

Use the Multiplication Property � ��13

� � 6�(w3 � 2)(x7)(y1 � 5)of Exponents.

Simplify. � 2w5x7y6

Example 6.2.9

Example 6.2.8


Concept 2 has sections on

• Multiplying a Monomial bya Monomial

• Multiplying a Polynomialby a Monomial

• Dividing a Monomial by a Monomial

• Dividing a Polynomial by a Monomial

Multiplication Property of Exponents:xm � xn � xm � n

Find: (�5x3y)(3x5)(2xy5)

Solution

Write the coefficients next to � (�5x3y)(3x5)(2xy5)each other. Write the factors with base xnext to each other and write � (�5 � 3 � 2)(x3 � x5 � x1)(y1 � y5)the factors with base y next to each other.

Use the Multiplication Property � (�5 � 3 � 2)(x3 � 5 � 1)(y1 � 5)of Exponents.

Simplify. � �30x9y6

Multiplying a Polynomial By a MonomialTo multiply a monomial by a polynomial with more than one term, use the Distributive Property to distribute the monomial to each term in the polynomial.

Find: �8w3y(4w2y5 � w4)

Solution � �8w3y(4w2y5 � w4)

Multiply each term in the polynomial by the � (�8w3y)(4w2y5) � (�8w3y)(w4)monomial, �8w3y.

Within each term, write the coefficients next to each other. Write thefactors with base w next to each other and write the factors with base ynext to each other.

� (�8 � 4)(w3 � w2)(y � y5) � (�8)(w3 � w4)(y)

Use the Multiplication � (�8 � 4)(w3 � 2y1 � 5) � (�8)(w3 � 4y)Property of Exponents.

Simplify. � �32w5y6 � 8w7y

Example 6.2.11

Example 6.2.10


Find: 5x4(3x2y2 � 2xy2 � x3y)

Solution � 5x4(3x2y2 � 2xy2 � x3y)

Multiply each term in the polynomial by the monomial, 5x4.

� (5x4)(3x2y2) � (5x4)(2xy2) � (5x4)(x3y)

Within each term, write the coefficients next to each other. Write thefactors with base x next to each other and write the factors with base y nextto each other.

� (5 � 3)(x4x2y2) � (5 � 2)(x4x1y2) � (5 � 1)(x4x3y)

Use the Multiplication Property of Exponents.

� (5 � 3)(x4 � 2y2) � (5 � 2)(x4 � 1y2) � (5 � 1)(x4 � 3y)

Simplify. � 15x6y2 � 10x5y2 � 5x7y

Dividing a Monomial By a MonomialTo divide a monomial by a monomial, use the Division Property ofExponents. (Assume that any variable in the denominator is not equal to zero.)

Division Property of Exponents

�xx

m

n� � xm � n for m � n and x � 0

�xx

m

n� � �xn

1� m� for m � n and x � 0

Find: �36w5xy3 9w2y7

Solution � �36w5xy3 9w2y7

Rewrite the problem using a � ��3

96ww2y

5

7xy3

�

division bar.

Cancel the common factor, 9, � ��4

ww2y

5

7xy3

�

in the numerator and denominator.

Use the Division Property of Exponents. � ��4

yw7 �

5 �

3

2x�

Simplify. � ��4

yw4

3x�

Example 6.2.13

Example 6.2.12


Dividing a Polynomial By a Monomial

When you added fractions, you learned: �ac

� � �bc

� � �a �

cb

�, where c � 0

If we exchange the expressions on either side of the equals sign, we have: �

a �

cb

� � �ac

� � �bc

�

We will use this property to divide a polynomial by a monomial. To divide a polynomial by a monomial, divide each term of thepolynomial by the monomial.

Find: (27w5x3y2 � 12w3x2y) 3w2xy

Solution � (27w5x3y2 � 12w3x2y) 3w2xy

Rewrite the problem using a �division bar.

Divide each term of the � �27

3ww

5

2xx

3

yy2

� � �132ww

2

3

xxy

2y�

polynomial by the monomial.

Cancel the common factor, 3, � �9w

w

5

2xx

3

yy2

� � �4ww

2

3

xxy

2y�

in each fraction.

Use the Division Property ��9w5 � 2x

1

3 � 1y2 � 1��

4w3 � 2x1

2 � 1y1 � 1�

of Exponents.

Note that y1 � 1 � y0 � 1. � 9w3x2y � 4wx

A landscape architect is designing a patio. She wants to estimate the costof the patio for various widths and lengths.

a. Construct an expression for the area of the patio in terms of x and y.

b. If the brick she will use costs $4.50 per square foot, find the cost of the brick for a patio that is 10 feet wide by 40 feet long.

x � 2y

x � 2yy y

y y

length = x

width = y

Example 6.2.15

27w5x3y2 � 12w3x2y��

3w2xy

Example 6.2.14


Solution

a. The patio is made up of two triangles and a rectangle. Recall two formulas from geometry:

Area of a rectangle � length � width Area � lw

Area of a triangle � �12

�(base)(height) Area � �12

�bh

Express the area of each triangle Area � �12

�bhin terms of y.

Each triangle has base y � �12

�(y)(y)and height y.

� �12

�y2

Express the area of the rectangle Area � lwin terms of x and y.

The rectangle has length (x � 2y) � (x � 2y)yand width y. � xy � 2y2

The area of the patio is the sum of the areas of the two Area � � �triangles and the rectangle.

Substitute the expressions for area. � �12

�y2 � xy � 2y2 � �12

�y2

Simplify. � xy � y2

Therefore, the area of the patio in terms of x and y is xy � y2.

b. The length of the base of the patio is x. This is 40 feet. The width of the patio, y, is 10 feet.

In the formula for the area of Area � xy � y2

the patio, substitute 40 feet � (40 feet)(10 feet) � (10 feet)2

for x and 10 feet for y. � 300 feet2

The cost of the patio is the Cost � �1$f4o.5o0t24

� � 300 feet24

price per square foot times the number of square feet. � $1350

The bricks for the patio will cost $1350.

20 feet

20 feet

10 feet 10 feet

length = 40 feet

width = 10 feet

10 feet 10 feet

area oftriangle

area ofrectangle

area oftriangle


Here is a summary of this concept from Interactive Mathematics.


LESSON 6.2 POLYNOMIAL OPERATIONS I CHECKLIST 395

monomialpolynomialbinomialtrinomial

degree of a termdegree of a polynomialevaluate a polynomial

Ideas and Procedures❶ Definition of a Polynomial

Determine whether a given expression is a Example 6.2.1bpolynomial. Determine if �

2x42� � 3x is a polynomial.

See also: Example 6.2.1a, c, dApply 1-4

❷ Degree of a PolynomialArrange the terms of a polynomial in descending Example 6.2.2order by degree and determine the degree of the Arrange the terms of this polynomial in descending polynomial. order and determine the degree of the polynomial:

7x3 � 8 � 2x � x4

See also: Apply 5-7

❸ Evaluate a PolynomialEvaluate a polynomial when given a specific value Example 6.2.3for each variable. Evaluate this polynomial when w � �3 and y � 2:

6w2 � 4wy � y4 � 5

See also: Apply 8-13

❹ Add PolynomialsFind the sum of polynomials. Example 6.2.5

Find the sum of (3z3 � 2zy2 � 6y3) and (15z3 � 5zy2 � 4z2).

See also: Example 6.2.4Apply 14-22

❺ Subtract PolynomialsFind the difference of polynomials. Example 6.2.7

Subtract (15z2 � 5yz2 � 4y3) from (6y3 � 10z3 � 2yz2).


Checklist Lesson 6.2Here is what you should know after completing this lesson.

Words and Phrases

❻ Multiply MonomialsFind the product of monomials. Example 6.2.10

Find: (�5x3y)(3x5)(2xy5)

See also: Example 6.2.8, 6.2.9Apply 29-35

❼ Multiply a Monomial by a PolynomialFind the product of a monomial and a polynomial. Example 6.2.12

Find: 5x4(3x2y2 � 2xy2 � x3y)


❽ Divide a Monomial by a MonomialFind the quotient of two monomials. Example 6.2.13

Find: �36w5xy3 9w2y7

See also: Apply 42-50

❾ Divide a Polynomial by a MonomialFind the quotient of a polynomial divided by a Example 6.2.14monomial. Find: (27w5x3y2 � 12w3x2y) 3w2xy



LESSON 6.2 POLYNOMIAL OPERATIONS I HOMEWORK 397

ExplainAdding and SubtractingPolynomials1. Circle the algebraic expression that is a

polynomial.

3�14

�y3 � �3y2 ��5�

3�14

�y3 � 3y2 � 5

�41y3� � 3y2 � 5

2. Write m beside the monomial, b beside thebinomial, and t beside the trinomial.

____ 34x � x2 � z

____ wxy3z2

____ pn2 � 13n3

3. Given the polynomial 3y � 2y3 � 4y5 � 2:

a. write the terms in descending order.

b. find the degree of each term.

c. find the degree of the polynomial.

4. Find: (�3w � 12w3 � 2) � (15w � 2w3 � 4w5 � 3)

5. Find: (2v3 � 6v2 � 2) � (5v � v3 � 4v7 � 3)

6. Evaluate �14

�xy � 3y2 � 5x3 when x � 2 and y � 4.

7. Find: (�s2t � s3t3 � 4st2 � 27) �(3st2 � 2st � 8s3t3 � 13t � 36)

8. Find: (12x3y � 9x2y2 � 6xy � y � 7) �(7xy � x � y � 11x3y � 3x2y2 � 4)

9. Angelina works at a pet store. Today, she iscleaning three fish tanks. These polynomialsdescribe the volumes of the tanks:

Tank 1: xy2

Tank 2: x2y � 2y3 � 4xy2 � 3

Tank 3: x2y � 5xy2 � 6y3

Write a polynomial that describes the total volumeof the three tanks. Hint: Add the polynomials.

volume � ________

10. Angelina has three fish tanks to clean. Thesepolynomials describe their volumes.

Tank 1: xy2

Tank 2: x2y � 2y3 � 4xy2 � 3

Tank 3: x2y � 5xy2 � 6y3

What is the total volume of the fish tanks if x � 3 feet and y � 1.5 feet?

volume � ________ cubic feet

11. Find: (w2yz � 3w3 � 2wyz2 � 4wyz) �(4wy2z � 3w2yz � 2wyz2) � (2wyz � 3)

12. Find: (tu2v � 4t2u2v � 9t3uv � 3tv) �(3t2u2 � 2tv � t3) � (4t2u2v � 3tv � 2tu2v)� (6t3uv � 2tv)

Multiplying and DividingPolynomials13. Find: xyz � x2y2z2

14. Find: 3p2r � 2p3qr

15. Find: �6t3u2v11 � �12

�tu2v4

16. Find: 3y(2x3 � 3x2y)

17. Find: 5p2r3(2pr � p2r2)

18. Find: t3uv4(2tu � 3uv � 4tv � 5)

19. Write 12w7x3y2z6 4w2x2y3z6 as a fraction andsimplify.

Homework

Homework Problems

Circle the homework problems assigned to you by the computer, then complete them below.


20. Write (36x3y3 � 15x2y5) 9x2y as a fraction and simplify.

21. Find: 15a7b4d2 10a4b9c3d

22. Tony is an algebra student. This is how heanswered a question on a test:(2t8u3 � 4t4u9 � 6t12u6) 2t4u3 �t2u � 2tu3 � 3t3u2

Is his answer right or wrong? Why? Circle the mostappropriate response.

The answer is right.

The answer is wrong. Tony divided theexponents rather than adding them. The correct answer is t12u6 � t8u12 � t16u9.

The answer is wrong. The terms need to be ordered by degree. The correct answer is 3t3u2 � t2u � 2tu3.

The answer is wrong. Tony divided theexponents rather than subtracting them. The correct answer is t4 � 2u6 � 3t8u3.

The answer is wrong. Tony shouldn't havecanceled the numerical coefficients. The correct answer is 2t2u � 4tu3 � 6t3u2.

23. Find: (16x2y4 � 20x3y5) 12xy2

24. Find: (20t5u11 � 5t3u5 � 30tu6v5) 10t4u5

LESSON 6.2 POLYNOMIAL OPERATIONS I APPLY 399

Adding and SubtractingPolynomials1. Circle the algebraic expressions below that are

polynomials.

2xy � 5xz

�32x� � 6x

9y2 � 13yz � 8z2

�24x5�

�155aa8

3�

2. Circle the algebraic expressions below that arepolynomials.

8xy � �3y

�

�17x3�3w � 7wz � 1

7x2 � 13x � 8y2

�132xx3

2�

3. Identify each polynomial below as a monomial, abinomial, or a trinomial.

a. 17x � 24z

b. 13ab2 � 5

c. m � n � 10

d. 42a2b4c

e. 73 � 65x � 21y

4. Identify each polynomial below as a monomial, abinomial, or a trinomial.

a. 25 � 6xyz � 4x

b. 2xyz3

c. x � y � 1

d. 36 � 3xyz

e. 32x2y

5. Find the degree of the polynomial 8a3b5 � 11a2b3 � 7b6.

6. Find the degree of the polynomial 12m4n7 � 16m12.

7. Find the degree of the polynomial 7x3y2z � 3x2y3z4 � 6z7.

8. Evaluate 2x2 � 8x � 11 when x � �1.

9. Evaluate x3 � 3x2 � x � 1 when x � �2.

10. Evaluate 2x2 � 5x � 8 when x � 3.

11. Evaluate x2y � xy2 when x � 2 and y � �3.

12. Evaluate 5mn � 4mn2 � 8m � n when m � 4 andn � �2.

13. Evaluate 3uv � 6u2v � 2u � v � 4 when u � 2and v � �4.

14. Find: (3x2 � 7x) � (x2 � 5)

15. Find: (5x2 � 4x � 8) � (x2 � 7x)

16. Find: (6a2 � 8a � 10) � (�3a2 � 2a � 7)

17. Find: (12m2n3 � 7m2n2 � 14mn) �(3m2n3 � 5m2n2 � 7mn)

18. Find: (10x4y3 � 9x2y3 � 6xy2 � x) �(28x4y3 � 14x2y3 � 3xy2 � x)

19. Find: (13a3b2 � 6a2b � 5ab3 � b) �(2a3b2 � 2a2b � 4ab3 � b)

20. Find: (11u5v4w3 � 6u3v2w) �(6u5v4w3 � 11u3v2w)

21. Find: (7xy2z3 � 19x2yz2 � 26x3y3z) �(13xy2z3 � 11x2yz2 � 16x3y3z)

22. Find: (9a4b2c � 3a2b3c � 5abc) �(2abc � 6a4b2c � 2) � (3a2b3c � 5)

23. Find: (5x3 � 7x) � (x3 � 8)

24. Find: (9a2 � 7ab � 14b) � (3a2 � 7b)

Apply

Practice Problems

Here are some additional practice problems for you to try.


25. Find: (2y2 � 6xy � 3y) � (y2 � y)

26. Find: (8x3 � 9x2 � 17) � (5x3 � 3x2 � 15)

27. Find: (9a5b3 � 8a4b � 6b) �(�2a5b3 � 12a4b � 3b)

28. Find: (7x4y2 � 3x2y � 5x) � (9x4y2 � 3x2y � 2x)

Multiplying and DividingPolynomials29. Find: 3y4 � 5y

30. Find: 5x3 � 2x

31. Find: �5a5 � 9a4

32. Find: �3x3 � 12x4

33. Find: 4x3y5 � 7xy3

34. Find: �7a5b6c3 � 8ab3c

35. Find: �3w2x3y2z � 2x2yz2

36. Find: 4y3(3y2 � 5y � 10)

37. Find: �2a3b2(3a4b5 � 5ab3 � 6a)

38. Find: 2xy3(2x6 � 5x4 � y2)

39. Find: 5a2b2(4a2 � 2a2b � 7ab2 � 3b)

40. Find: �4mn3(�3m2n � 12mn2 � 6m � 7n2)

41. Find: 4x3y3(3x3 � 7xy2 � 2xy � y)

42. Find: �93xx

3

2y

�

43. Find: �240aa

3

5

bb6

�

44. Find: �132xx

2

4

yy6

�

45. Find: �1322aa5

7

bb6

9

cc2�

46. Find: �1150mn

6

4np

1

3

0�

47. Find: �2146xw

6

xy

3

2

zz2

7�

48. Find: �271a5

4

abc

3

7cd

1

3

2d�

49. Find: �4228mmn

2

6

npq

3q5

4�

50. Find: �3261ww

2

5xy

3

2yz

7

2z

�

51. Find: �32a3

8�

a224a5

�

52. Find: �21m2

3�

mn18mn3�

53. Find: �14x �

2xy8x4y2�

54. Find: �24a2b1

2c6

3

a�

bc34ab4c5

�

55. Find: �32x2y1

3

6zx

4

3�

y3z84x5yz7

�

56. Find: �32r4s1t2

2

r�3s2

3tr2st5

�

1. Circle the expressions that are polynomials.

��325� �25

�p3r � 3p2q � �2r�

t2 � s � 5 �57

�c15 � �134�c11 � 3

m5n4o3p2r x2 � 3xy � �32x� � y2

2. Write m beside the monomial(s), b beside thebinomial(s), and t beside the trinomial(s).

a. ___ w5x4

b. ___ 2x2 � 36

c. ___ �13

�x17 � �23

�x12 � �13

�

d. ___ 27

e. ___ 27x3 � 2x2y3

f. ___ x2 � 3xy � �23

�y2

3. Given the polynomial 3w3 � 13w2 � 7w5 � 8w8 � 2, write the terms indescending order by degree.

4. Find:

a. (5x3y � 8x2y2 � 3xy � y3 � 13) � (�2xy � 6 � y2 � 4y3 � 2x3y)

b. (5x3y � 8x2y2 � 3xy � y3 � 13) � (�2xy � 6 � y2 � 4y3 � 2x3y)

5. Find: x3y2w � x5yw4

6. Find: n2p3(3n � 2n3p2 � 35p4)

7. Find: 21x5y2z7 14xyz

8. Find: (15t3u2v � 5t5uv2) 10tuv2

Evaluate

Practice Test

Take this practice test to be sure that you are prepared for the final quiz in Evaluate.

LESSON 6.2 POLYNOMIAL OPERATIONS I EVALUATE 401

Lesson 6.1 ExponentsHomework1a. 37 b. 57 c. 77 3a. 76 b. 76

5a. x 8 b. a8 c. 1 d. 7a. b 18 b. y 11 c.

9. 11a. 1 b. c. d. 3

Apply - Practice Problems1. 78 3. b 15 5. a 11 7. 96 9. n 5

11. 512 13. 1330 15. z 48 17. 81a 4

19. 8y 3 21. 23. 1 25. 1 27. 3

Evaluate - Practice Test1a. 114 b. 32y 5 c. 543 d. x 21y 26 e. 718 b14

2a. 23 b. b6 c. d.

3. and 4. (31x 8)0 � 5y, , and

5a. b48 b. 310a12 c. 299 x 44y 66

6a. b.

7a. –1 b. 1 c. –3 d. 1

8a. a 35 b.

Lesson 6.2 Polynomial Operations IHomework

1. 3 y 3 + 3y 2 – 5 3a. –4y 5 – 2y 3 + 3y + 2

b. 5, 3, 1, 0 c. 5 5. –4v7 + v 3 + 6v2 – 5v + 5

7. –7s3t 3 + 7st 2 – s2t + 2st –13t + 9

9. 2x 2y + 10xy 2 + 4y 3 + 3

11. 4w 2yz + 3w 3 – 4wyz 2 + 6wyz – 4wy 2z + 3

13. x 3y 3z 3 15. –3t 4u 4v 15 17. 10p 3r 4 + 5p4r 5

19. 21. 23. + or (4 + 5xy )xy 2�

35x 2y 3�

34xy 2�

33a 3d�2b 5c 3

3xw 5�

y

1�4

1�a35

76a6b24�

5654y 40�34x 32

(5y )2�

5y5y 2�

yx7y�x 4y 6

x 6y 2�x 3y 7

1�y 4

33�x9

m 4�n 6

1�b4

1�y 3

1�x

b3�a7

x3�

4

LESSON 6.2: ANSWERS 727

Apply - Practice Problems1. 2xy � 5xz ; 9y 2 � 13yz – 8z 2

3a.binomial b. binomial c. trinomial d. monomial

3e. trinomial

5. 8 7. 9 9. 7 11. 6 13. 84 15. 6x 2 � 11x – 8

17. 15m 2n 3 � 2m 2n 2 – 7mn 19. 15a 3b 2 � 4a 2b – ab 3

21. 20xy 2z 3 – 30x 2yz 2 � 10x 3y 3z 23. 4x 3 � 7x – 8

25. y 2 � 6xy � 4y 27. 11a 5b 3 – 4a 4b – 9b 29. 15y 5

31. –45a 9 33. 28x 4y 8 35. –6w 2x 5y 3z 3

37. –6a 7b 7 � 10a 4b 5 – 12a 4b 2

39. 20a 4b 2 � 10a 4b 3 – 35a 3b 4 – 15a 2b 3

41. 12x 6y 3 – 28x 4y 5 � 8x 4y 4 – 4x 3y 4

43. 5a 2b 5 45. 47. 49.

51. 4a � 3a 3 53. � 4x 3y 55. –

Evaluate - Practice Test 1. t 2 – s + 5, m5n4o3p2r, and c15 + c11 – 3�

2. w 5x 4 is a monomial.

2x 2 – 36 is a binomial.

x17 + x12 – is a trinomial.

27 is a monomial.

27x 3 – 2x 2y 3 is a binomial.

x 2 + 3xy – y 2 is a trinomial.

3. 8w 8 + 7w 5 + 3w 3 – 13w 2 – 2

4a. 3x 3y – 8x 2y 2 – 5y 3 + xy + y 2 + 19

b. 7x 3y – 8x 2y 2 + 3y 3 + 5xy – y 2 + 7

5. x 8y 3w 5

6. 3n3p3 + 2n5p5 – 35n2p 7

7.

8. – t 41�2

3t 2u�

2v

3x 4yz 6�

2

2�3

1�3

2�3

1�3

3�14

5�7

x 2z 3�2y 2

2�x

7�y

3n 5p 3�

2mq3x 3y 2z 5�

2w8a 2b 3�

3c

lesson 6.2 polynomial operations iansw.pdf · example 6.2.6 3z3 15z3 2zy2 5zy2 example 6.2.5 5x3...

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