copyright © 2013, 2009, 2006 pearson education, inc. 1 section 5.4 polynomials in several variables...
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Copyright © 2013, 2009, 2006 Pearson Education, Inc. 1
Section 5.4
Polynomials inSeveral
Variables
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Objective #1 Evaluate polynomials in several variables.
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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
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1. Substitute the given value for each variable.
2. Perform the resulting computation using the order of operations.
Evaluating a Polynomial in Several Variables
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Evaluate 5x3y + 6xy − 2x2y for x = 3 and y = −1.
1. Substitute the given value for each variable.
2. Perform the resulting computation using the order of operations.
)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
Evaluating a Polynomial in Several Variables
EXAMPLEEXAMPLE
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Evaluate 5x3y + 6xy − 2x2y for x = 3 and y = −1.
1. Substitute the given value for each variable.
2. Perform the resulting computation using the order of operations.
)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
Evaluating a Polynomial in Several Variables
EXAMPLEEXAMPLE
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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
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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
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Objective #2 Understand the vocabulary of polynomials
in two variables.
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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
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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
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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
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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 #2: Example
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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 #2: Example
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Objective #3 Add and subtract polynomials in several variables.
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• 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
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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
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Objective #3: Example
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
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Objective #3: Example
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
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Objective #3: Example
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
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Objective #3: Example
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
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Objective #4 Multiply polynomials in several variables.
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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.
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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.
Multiplying Polynomials in Several Variables
EXAMPLEEXAMPLE
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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.
Multiplying Polynomials in Several Variables
EXAMPLEEXAMPLE
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Objective #4: Example
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
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Objective #4: Example
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
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Objective #4: Example
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
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Objective #4: Example
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
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Objective #4: Example
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
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Objective #4: Example
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
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Objective #4: Example
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
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Objective #4: Example
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