copyright © 2013, 2009, 2006 pearson education, inc. 1 section 5.4 polynomials in several variables...

Copyright © 2013, 2009, 2006 Pearson Education, Inc. 1

Section 5.4

Polynomials inSeveral

Variables



Objective #1 Evaluate polynomials in several variables.


A polynomial containing two or more variables is called a polynomial in several variables. An example of a polynomial in two variables is:

yxxyyx 23 265

Polynomials in Several Variables


1. Substitute the given value for each variable.

2. Perform the resulting computation using the order of operations.

Evaluating a Polynomial in Several Variables


Evaluate 5x3y + 6xy − 2x2y for x = 3 and y = −1.



)1()3(2)1)(3(6)1()3(5 23

135

1818135

)1)(9(2)1)(3(6)1)(27(5

)1()3(2)1)(3(6)1()3(5 23


EXAMPLEEXAMPLE


Evaluate 5x3y + 6xy − 2x2y for x = 3 and y = −1.



)1()3(2)1)(3(6)1()3(5 23

3 25(3) ( 1) 6(3)( 1) 2(3) ( 1)

5(27)( 1) 6(3)( 1) 2(9)( 1)

135 18 18

135


EXAMPLEEXAMPLE


Objective #1: Example

1. Evaluate 3 23 5 6x y xy y for 1x and 5.y

Begin by substituting 1 in for x and 5 in for y.

3 2 3 23 5 6 3( 1) (5) ( 1)(5) 5(5) 6

3( 1)(5) ( 1)(25) 5(5) 6

15 25 25 6

9

x y xy y



1. Evaluate 3 23 5 6x y xy y for 1x and 5.y

Begin by substituting 1 in for x and 5 in for y.

3 2 3 23 5 6 3( 1) (5) ( 1)(5) 5(5) 6

3( 1)(5) ( 1)(25) 5(5) 6

15 25 25 6

9

x y xy y


Objective #2 Understand the vocabulary of polynomials

in two variables.


Polynomials

In general, a polynomial in two variables, x and y, contains the sum of one or more monomials in the form The constant, a, is the coefficient. The exponents, n and m, represent whole numbers. The degree of the monomial is n + m.

.n max y

n max y


Polynomials

EXAMPLEEXAMPLE

SOLUTIONSOLUTION

Determine the coefficient of each term, the degree of each term, the degree of the polynomial, the leading term, and the leading coefficient of the polynomial.

735 yx

4512 2734 xyxyx

2x

yx412


Polynomials

CONTINUEDCONTINUED

The degree of the polynomial is the greatest degree of all its terms, which is 10. The leading term is the term of the greatest degree, which is . Its coefficient, −5, is the leading coefficient.

735 yx

4512 2734 xyxyx


2. Determine the coefficient of each term, the degree of each term, and the degree of the polynomial:

4 5 3 2 28 7 5 11x y x y x y x

Term Term Term Term Term

4 5 3 2 28 7 5 11x y x y x y x

4 5

3 2

2

Term Coefficient Degree

8 8 4 5 9

7 7 3 2 5

1 2 1 3

5 5 1

11 11 0

x y

x y

x y

x

The degree of the polynomial is the highest degree of all its terms, which is 9.



Objective #3 Add and subtract polynomials in several variables.


• Polynomials in several variables are added by combining like terms.

• Polynomials in several variables are subtracted by adding the first polynomial and the opposite of the second polynomial.

Like terms are terms containing exactly the same variables to the same powers.

Adding and Subtracting Polynomials in Several Variables


Subtracting Polynomials

EXAMPLEEXAMPLE

SOLUTIONSOLUTION

Subtract . 8653765 324324 xyyxyxyyxyx

xyyxyxyyxyx 8653765 324324 4 2 3 4 2 35 6 7 3 5 6 8x y x y y x y x y y x Change subtraction to

addition and change the sign of every term of the polynomial in parentheses.

Rearrange terms

Combine like terms

4 2 4 2 3 3

4 2 3

5 3 6 5 7 6 8

= 2 11 8

x y x y x y x y y y x

x y x y y x



3a. Add: 2 2( 8 3 6) (10 5 10)x y xy x y xy

2 2

2 2

2 2

2

( 8 3 6) (10 5 10)

8 3 6 10 5 10

8 10 3 5 6 10

2 2 4

x y xy x y xy

x y xy x y xy

x y x y xy xy

x y xy



3b. Subtract: 3 2 2 3 2 2(7 10 2 5) (4 12 3 5)x x y xy x x y xy

3 2 2 3 2 2

3 2 2 3 2 2

3 3 2 2 2 2

3 2 2

(7 10 2 5) (4 12 3 5)

7 10 2 5 4 12 3 5

7 4 10 12 2 3 5 5

3 2 5 10

x x y xy x x y xy

x x y xy x x y xy

x x x y x y xy xy

x x y xy


Objective #4 Multiply polynomials in several variables.


Multiplying Polynomials in Several Variables

The product of monomials forms the basis of polynomial multiplication. As with monomials in one variable, multiplication can be done mentally by multiplying coefficients and adding exponents on variables with the same base.


Multiply coefficients and add exponents on variables with the same base.

74

523

523

28

))()(74(

)7)(4(

yx

yyxx

xyyx

Regroup.

Multiply the coefficients and add the exponents.


EXAMPLEEXAMPLE


Multiply each term of the polynomial by the monomial.

yxyxyx

yxxyyxyxyx

xyyxyx

32437

33243

243

8288

2)4()7)(4()2)(4(

)272)(4(

Use the distributive property.

Multiply the coefficients and add the exponents.


EXAMPLEEXAMPLE



4a. Multiply: 3 4 2(6 )(10 )xy x y

3 4 2 4 3 2

1 4 3 2

5 5

(6 )(10 ) (6 10)( )( )

60

60

xy x y x x y y

x y

x y



4b. Multiply: 2 4 5 26 (10 2 3)xy x y x y

2 4 5 2

2 4 5 2 2 2

1 4 2 5 1 2 2 1 2

5 7 3 3 2

6 (10 2 3)

6 10 6 2 6 3

60 12 18

60 12 18

xy x y x y

xy x y xy x y xy

x y x y xy

x y x y xy



4c. Multiply: (7 6 )(3 )x y x y

OF I L

2 2

2 2

(7 6 )(3 )

(7 )(3 ) (7 )( ) ( 6 )(3 ) ( 6 )( )

21 7 18 6

21 25 6

x y x y

x x x y y x y y

x xy xy y

x xy y



4d. Multiply: 2 2(6 5 )(6 5 )xy x xy x

2 2( )( )

2 2 2 2 2

2 4 2

(6 5 )(6 5 ) (6 ) (5 )

36 25

A B A B A B

xy x xy x xy x

x y x

copyright © 2013, 2009, 2006 pearson education, inc. 1 section 5.4 polynomials in several variables...

Documents