Slide 5- 1Copyright © 2012 Pearson Education, Inc.

Copyright © 2012 Pearson Education, Inc.

5.7 Polynomials in Several

Variables

■ Evaluating Polynomials

■ Like Terms and Degrees

■ Addition and Subtraction

■ Multiplication

■ Function Notation

Slide 5- 3Copyright © 2012 Pearson Education, Inc.

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

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

Example

Slide 5- 4Copyright © 2012 Pearson Education, Inc.

The 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

Example

h

r

Slide 5- 5Copyright © 2012 Pearson Education, Inc.

continued h = 50 ft and r = 9 ft

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).

Slide 5- 6Copyright © 2012 Pearson Education, Inc.

Recall that the degree of a monomial is the number of variable factors in the term.Example Identify the coefficient and the degree of each term and the degree of the polynomial

10x3y2 15xy3z4 + yz + 5y + 3x2 + 9.

Term Coefficient Degree Degree of the Polynomial

10x3y2 10 5

815xy3z4 15 8

yz 1 2

5y 5 1

3x2 3 2

9 9 0

Slide 5- 7Copyright © 2012 Pearson Education, Inc.

Like Terms

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.

Slide 5- 8Copyright © 2012 Pearson Education, Inc.

a) 10x2y + 4xy3 6x2y 2xy3

b) 8st 6st2 + 4st2 + 7s3 + 10st 12s3 + t 2Solution a) 10x2y + 4xy3 6x2y 2xy3

= (10 6)x2y + (42)xy3

= 4x2y + 2xy3

b) 8st 6st2 + 4st2 + 7s3 + 10st 12s3 + t 2= 5s3 2st2 + 18st + t 2

Example Combine like terms.

Slide 5- 9Copyright © 2012 Pearson Education, Inc.

Addition and Subtraction

Example Add: (6x3 + 4y 6y2) + (7x3 + 5x2 + 8y2).

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

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

= x3 + 5x2 + 4y + 2y2

Slide 5- 10Copyright © 2012 Pearson Education, Inc.

Subtract:

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

Solution

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

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

= 9x3y2 + 4x2y3 x2y2 + 13y

Example

Slide 5- 11Copyright © 2012 Pearson Education, Inc.

Multiplication

Example Multiply: (4x2y 3xy + 4y)(xy + 3y).

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

12x2y2 9xy2 + 12y2

4x3y2 3x2y2 + 4xy2

4x3y2 + 9x2y2 5xy2 + 12y2

Slide 5- 12Copyright © 2012 Pearson Education, Inc.

Multiply.a) (x + 6y)(2x 3y) b) (5x + 7y)2

c) (a4 5a2b2)2 d) (7a2b + 3b)(7a2b 3b)e) (3x3y2 + 7t)(3x3y2 + 7t)f) (3x + 1 4y)(3x + 1 + 4y)

Solution a) (x + 6y)(2x 3y) = 2x2 3xy + 12xy 18y2

= 2x2 + 9xy 18y2 FOIL

Example

Slide 5- 13Copyright © 2012 Pearson Education, Inc.

b) (5x + 7y)2 = (5x)2 + 2(5x)(7y) + (7y)2

= 25x2 + 70xy + 49y2

c) (a4 5a2b2)2 = (a4)2 2(a4)(5a2b2) + (5a2b2)2

= a8 10a6b2 + 25a4b4

d) (7a2b + 3b)(7a2b 3b) = (7a2b)2 (3b)2

= 49a4b2 9b2

Solution continued

Slide 5- 14Copyright © 2012 Pearson Education, Inc.

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

= 49t2 9x6y4

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

= 9x2 + 6x + 1 16y2

Solution continued

Slide 5- 15Copyright © 2012 Pearson Education, Inc.

Function Notation

Example Given f(x) = x2 – 3x + 2, find and simplify f(a + 5).

SolutionTo find f(a + 5), we replace x with a + 5 and simplify.

f(a + 5) = (a + 5)2 – 3 (a + 5) + 2 = a2 + 10a + 25 – 3a – 15 + 2

= a2 + 7a + 12