CHAPTER

4Polynomials: Operations

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4.1 Integers as Exponents

4.2 Exponents and Scientific Notation

4.3 Introduction to Polynomials

4.4 Addition and Subtraction of Polynomials

4.5 Multiplication of Polynomials

4.6 Special Products

4.7 Operations with Polynomials in Several Variables

4.8 Division of Polynomials

OBJECTIVES

4.7 Operations with Polynomials in Several Variables

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a Evaluate a polynomial in several variables for given values of the variables.

b Identify the coefficients and the degrees of the terms of a polynomial and the degree of a polynomial.

c Collect like terms of a polynomial.d Add polynomials.e Subtract polynomials.f Multiply polynomials.

EXAMPLE

Solution We substitute 3 for x and 4 for y:

5 + 4x + xy2 + 9x3y2 = 5 + 4(3) + (3)(42) + 9(3)3(4)2

= 5 12 48 3888= 3943

4.7 Operations with Polynomials in Several Variables

a Evaluate a polynomial in several variables for given values of the variables.

A Evaluate the polynomial 5 + 4x + xy2 + 9x3y2 for x = 3 and y = 4.

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EXAMPLEThe surface area of a right circular cylinder is given by the polynomial 2rh + 2r2 where h is the height and r is the radius of the base. A barn silo has a height of 50 feet and a radius of 9 feet. Approximate its surface area.Solution We evaluate the polynomial for h = 50 ft and r = 9 ft. If 3.14 is used to approximate , we have

h

r

4.7 Operations with Polynomials in Several Variables

a Evaluate a polynomial in several variables for given values of the variables.

B Applications of Polynomials

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EXAMPLE

h = 50 ft and r = 9 ft

4.7 Operations with Polynomials in Several Variables

a Evaluate a polynomial in several variables for given values of the variables.

A Applications of Polynomials

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2rh + 2r2 2(3.14)(9 ft)(50 ft) + 2(3.14)(9 ft)2

2(3.14)(9 ft)(50 ft) + 2(3.14)(81 ft2) 2826 ft2 + 508.68 ft2 3334.68 ft2

Note that the unit in the answer (square feet) is a unit of area. The surface area is about 3334.7 ft2 (square feet).

Recall that the degree of a monomial is the number of variable factors in the term.

4.7 Operations with Polynomials in Several Variables

b Identify the coefficients and the degrees of the terms of a polynomial and the degree of a polynomial.

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Example Identify the coefficient and the degree of each term and the degree of the polynomial:

10x3y2 – 15xy3z4 + yz + 5y + 3x2 + 9

4.7 Operations with Polynomials in Several Variables

b Identify the coefficients and the degrees of the terms of a polynomial and the degree of a polynomial.

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Term Coefficient Degree Degree of the Polynomial10x3y2

15xy3z4

yz

5y

3x2

9

10x3y2 – 15xy3z4 + yz + 5y + 3x2 + 9

4.7 Operations with Polynomials in Several Variables

b Identify the coefficients and the degrees of the terms of a polynomial and the degree of a polynomial.

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Like, or similar terms either have exactly the same variables with exactly the same exponents or are constants.For example,

9w5y4 and 15w5y4 are like termsand

–12 and 14 are like terms,but

–6x2y and 9xy3 are not like terms.

4.7 Operations with Polynomials in Several Variables

c Collect like terms of a polynomial.

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EXAMPLEa) 10x2y + 4xy3 6x2y 2xy3

= (10 6)x2y + (42)xy3

= 4x2y + 2xy3

4.7 Operations with Polynomials in Several Variables

c Collect like terms of a polynomial.

C Combine like terms.

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EXAMPLEa) 10x2y + 4xy3 6x2y 2xy3

4.7 Operations with Polynomials in Several Variables

c Collect like terms of a polynomial.

C Combine like terms.

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EXAMPLE

b) 8st 6st2 + 4st2 + 7s3 + 10st 12s3 + t 2

4.7 Operations with Polynomials in Several Variables

c Collect like terms of a polynomial.

C Combine like terms.

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EXAMPLE

Solution(–6x3 + 4y – 6y2) + (7x3 + 5x2 + 8y2)

= x3 + 5x2 + 4y + 2y2

4.7 Operations with Polynomials in Several Variables

d Add polynomials.

D Add: (–6x3 + 4y – 6y2) + (7x3 + 5x2 + 8y2)

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EXAMPLE(5x2y + 2x3y2 + 4x2y3 + 7y) (5x2y 7x3y2 + x2y2 6y)

Solution(5x2y + 2x3y2 + 4x2y3 + 7y) (5x2y 7x3y2 + x2y2 6y) = = 9x3y2 + 4x2y3 x2y2 + 13y

4.7 Operations with Polynomials in Several Variables

e Subtract polynomials.

E Subtract:

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EXAMPLE

4.7 Operations with Polynomials in Several Variables

f Multiply polynomials.

F Multiply: (4x2y 3xy + 4y)(xy + 3y)

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Solution 4x2y 3xy + 4y xy + 3y

12x2y2 9xy2 + 12y2

4x3y2 3x2y2 + 4xy2

4x3y2 + 9x2y2 5xy2 + 12y2

EXAMPLE

4.7 Operations with Polynomials in Several Variables

f Multiply polynomials.

F Multiply:

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(4x2y 3xy + 4y)(xy + 3y)

4.7 Operations with Polynomials in Several Variables

f Multiply polynomials.

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The special products discussed in Section 4.5 can speed up your work.

EXAMPLE

4.7 Operations with Polynomials in Several Variables

f Multiply polynomials.

G Multiply.

(continued)

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a) (x + 6y)(2x 3y)

Solution

= 2x2 3xy + 12xy 18y2

= 2x2 + 9xy 18y2

FOIL

EXAMPLE

b) (5x + 7y)2 =

c) (a4 5a2b2)2 =

4.7 Operations with Polynomials in Several Variables

f Multiply polynomials.

G Multiply.

(continued)

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EXAMPLE

d) (7a2b + 3b)(7a2b 3b) =

4.7 Operations with Polynomials in Several Variables

f Multiply polynomials.

G Multiply.

(continued)

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EXAMPLEe) (3x3y2 + 7t)(3x3y2 + 7t)

= (7t 3x3y2)(7t + 3x3y2) = (7t)2 (3x3y2)2

= 49t2 9x6y4

4.7 Operations with Polynomials in Several Variables

f Multiply polynomials.

G Multiply.

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EXAMPLE

f) (3x + 1 4y)(3x + 1 + 4y) ((3x+1) – 4y)((3x+1) + 4y)

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

= 9x2 + 6x + 1 16y2

4.7 Operations with Polynomials in Several Variables

f Multiply polynomials.

G Multiply.

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