lesson 6.2 polynomial operations iansw.pdf · example 6.2.6 3z3 15z3 2zy2 5zy2 example 6.2.5 5x3...
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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
LESSON 6.2 POLYNOMIAL OPERATIONS I EXPLAIN 385
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
386 TOPIC 6 EXPONENTS AND POLYNOMIALS
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
LESSON 6.2 POLYNOMIAL OPERATIONS I EXPLAIN 387
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
388 TOPIC 6 EXPONENTS AND POLYNOMIALS
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
LESSON 6.2 POLYNOMIAL OPERATIONS I EXPLAIN 389
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
390 TOPIC 6 EXPONENTS AND POLYNOMIALS
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
LESSON 6.2 POLYNOMIAL OPERATIONS I EXPLAIN 391
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 $4.50 per square foot, find the cost of the brick for a patio that is 10 feet wide by 40 feet long.
x � 2y
x � 2yy y
y y
length = x
width = y
Example 6.2.15
27w5x3y2 � 12w3x2y���
3w2xy
Example 6.2.14
392 TOPIC 6 EXPONENTS AND POLYNOMIALS
Solution
a. The patio is made up of two triangles and a rectangle. Recall two formulas from geometry:
Area of a rectangle � length � width Area � lw
Area of a triangle � �12
�(base)(height) Area � �12
�bh
Express the area of each triangle Area � �12
�bhin terms of y.
Each triangle has base y � �12
�(y)(y)and height y.
� �12
�y2
Express the area of the rectangle Area � lwin terms of x and y.
The rectangle has length (x � 2y) � (x � 2y)yand width y. � xy � 2y2
The area of the patio is the sum of the areas of the two Area � � �triangles and the rectangle.
Substitute the expressions for area. � �12
�y2 � xy � 2y2 � �12
�y2
Simplify. � xy � y2
Therefore, the area of the patio in terms of x and y is xy � y2.
b. The length of the base of the patio is x. This is 40 feet. The width of the patio, y, is 10 feet.
In the formula for the area of Area � xy � y2
the patio, substitute 40 feet � (40 feet)(10 feet) � (10 feet)2
for x and 10 feet for y. � 300 feet2
The cost of the patio is the Cost � �1$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
LESSON 6.2 POLYNOMIAL OPERATIONS I EXPLAIN 393
Here is a summary of this concept from Interactive Mathematics.
394 TOPIC 6 EXPONENTS AND POLYNOMIALS
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).
See also: Example 6.2.6Apply 23-28
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)
See also: Example 6.2.11Apply 36-41
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
See also: Example 6.2.15Apply 51-56
396 TOPIC 6 EXPONENTS AND POLYNOMIALS
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.
398 TOPIC 6 EXPONENTS AND POLYNOMIALS
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.
400 TOPIC 6 EXPONENTS AND POLYNOMIALS
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
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