Transcript 0371-5.4
Section 5.4
Polynomials in Several Variables
Objective #1
Evaluate polynomials in several variables. Copyright © 2013, 2009, 2006 Pearson Education, Inc.
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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: 5
x
3
y
6
xy
2
x
2
y
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Evaluating a Polynomial in Several Variables 1. Substitute the given value for each variable.
2. Perform the resulting computation using the order of operations.
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Evaluating a Polynomial in Several Variables
EXAMPLE
1.
Evaluate 5
x
3
y
+ 6
xy
− 2
x
2
y
for
x
= 3 and
y
= −1.
Substitute the given value for each variable.
5 ( 3 ) 3 ( 1 ) 6 ( 3 )( 1 ) 2 ( 3 ) 2 ( 1 ) 2.
Perform the resulting computation using the order of operations.
5 ( 3 ) 3 ( 1 ) 6 ( 3 )( 1 ) 2 ( 3 ) 2 ( 1 ) 5 ( 27 )( 1 ) 6 ( 3 )( 1 ) 2 ( 9 )( 1 ) 135 18 18 135 Copyright © 2013, 2009, 2006 Pearson Education, Inc.
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Evaluating a Polynomial in Several Variables
EXAMPLE
1.
Evaluate 5
x
3
y
+ 6
xy
− 2
x
2
y
for
x
= 3 and
y
= −1.
Substitute the given value for each variable.
5 ( 3 ) 3 ( 1 ) 6 ( 3 )( 1 ) 2 ( 3 ) 2 ( 1 ) 2.
Perform the resulting computation using the order of operations.
2 135 Copyright © 2013, 2009, 2006 Pearson Education, Inc.
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Objective #1: Example
1.
Evaluate 3
xy
2 5
y
6 for
x
1 and
y
Begin by substituting 1 in for
x
and 5 in for
y
. 5.
3
xy
2 5
y
2 5(5) 6 6 9 Copyright © 2013, 2009, 2006 Pearson Education, Inc.
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Objective #1: Example
1.
Evaluate 3
xy
2 5
y
6 for
x
1 and
y
Begin by substituting 1 in for
x
and 5 in for
y
. 5.
3
xy
2 5
y
2 5(5) 6 6 9 Copyright © 2013, 2009, 2006 Pearson Education, Inc.
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Objective #2
Understand the vocabulary of polynomials in two variables. Copyright © 2013, 2009, 2006 Pearson Education, Inc.
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Polynomials In general, a
polynomial in two variables
,
x
and
y
, contains constant,
a
, is the
coefficient
. The exponents,
n
and
m
, represent whole numbers. The degree of the monomial
n ax y m
is
n
+
m
. Copyright © 2013, 2009, 2006 Pearson Education, Inc.
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Polynomials
EXAMPLE
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.
12
x
4
y
5
x
3
y
7
x
2 4
SOLUTION Term Coefficient
12
x
4
y
5
x
3
y
7
x
2 4 12 4
Degree (Sum of Exponents on the Variables)
4 + 1 = 5 3 + 7 = 10 2 + 0 = 2 0 + 0 = 0 Copyright © 2013, 2009, 2006 Pearson Education, Inc.
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Polynomials
CONTINUED
12
x
4
y
5
x
3
y
7
x
2 4 The degree of the polynomial is the greatest degree of all its terms, which is 10. The leading term is the term of the 5
x
3
y
7 − 5, is the leading coefficient.
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Objective #2: Example
2.
Determine the coefficient of each term, the degree of each term, and the degree of the polynomial: 8 7 5
x
11 8 Term Term 7 Term Term Term 5
x
11 Term Coefficient Degree 8 7 5
x
8 7 1 5 4 3 9 5 3 1 11 11 0 The degree of the polynomial is the highest degree of all its terms, which is 9. Copyright © 2013, 2009, 2006 Pearson Education, Inc.
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Objective #2: Example
2.
Determine the coefficient of each term, the degree of each term, and the degree of the polynomial: 8 7 5
x
11 8 Term Term 7 Term Term Term 5
x
11 Term Coefficient Degree 8 7 5
x
8 7 1 5 4 3 9 5 3 1 11 11 0 The degree of the polynomial is the highest degree of all its terms, which is 9. Copyright © 2013, 2009, 2006 Pearson Education, Inc.
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Objective #3
Add and subtract polynomials in several variables. Copyright © 2013, 2009, 2006 Pearson Education, Inc.
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Adding and Subtracting 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.
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Subtracting Polynomials
EXAMPLE
Subtract 5
x
4
y
2 6
x
3
y
7
y
3
x
4
y
2 5
x
3
y
6
y
8
x
.
SOLUTION
5
x
4
y
2 6
x
3
y
7
y
3
x
4
y
2 5
x
3
y
6
y
8
x
5 6 7
y
3 5 6
y
8
x
Change subtraction to addition and change the sign of every term of the polynomial in parentheses.
5 3 6 5 7
y
6
y
8
x
Rearrange terms = 2 11
x
Combine like terms Copyright © 2013, 2009, 2006 Pearson Education, Inc.
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Objective #3: Example
3a.
Add: 2 8 8 3
xy
5
xy
3
xy
3
xy
10 2
xy
4 (10 5
xy
10) 5
xy
10 3
xy
5
xy
10) Copyright © 2013, 2009, 2006 Pearson Education, Inc.
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Objective #3: Example
3a.
Add: 2 8 8 3
xy
5
xy
3
xy
3
xy
10 2
xy
4 (10 5
xy
10) 5
xy
10 3
xy
5
xy
10) Copyright © 2013, 2009, 2006 Pearson Education, Inc.
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Objective #3: Example
3b.
Subtract: (7
x
3 7
x
3 (7
x
10 3
x
3 2 3 10 10 7
x
3 4
x
3 2
xy
2 10 2 2
xy
2
xy
5
xy
2 2 12 10
x
3 12
x
3 12 3
xy
3
xy
2 5) 2 5)
x
3 12 3
xy
2 5 2
xy
2 3
xy
2 Copyright © 2013, 2009, 2006 Pearson Education, Inc.
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Objective #3: Example
3b.
Subtract: (7
x
3 7
x
3 (7
x
10 3
x
3 2 3 10 10 7
x
3 4
x
3 2
xy
2 10 2 2
xy
2
xy
5
xy
2 2 12 10
x
3 12
x
3 12 3
xy
3
xy
2 5) 2 5)
x
3 12 3
xy
2 5 2
xy
2 3
xy
2 Copyright © 2013, 2009, 2006 Pearson Education, Inc.
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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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Multiplying Polynomials in Several Variables
EXAMPLE
Multiply coefficients and add exponents on variables with the same base.
( 4
x
3
y
2 )( 7
xy
5 ) ( 4 7 )(
x
3
x
)(
y
2
y
5 ) Regroup.
28
x
4
y
7 Multiply the coefficients and add the exponents.
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Multiplying Polynomials in Several Variables
EXAMPLE
Multiply each term of the polynomial by the monomial.
( 4
x
3
y
)( 2
x
4
y
2 7
xy
2 ) ( 4
x
3
y
)( 2
x
4
y
2 ) 8
x
7
y
3 28
x
4
y
2 ( 4
x
3
y
)( 7
xy
) 8
x
3
y
( 4
x
3
y
) 2 Use the distributive property.
Multiply the coefficients and add the exponents.
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Objective #4: Example
4a.
Multiply: (6
xy
3 )(10 (6
xy
3 )(10 ) 60
x
60
y
) 4 )(
y
3
y
2 ) Copyright © 2013, 2009, 2006 Pearson Education, Inc.
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Objective #4: Example
4a.
Multiply: (6
xy
3 )(10 (6
xy
3 )(10 ) 60
x
60
y
) 4 )(
y
3
y
2 ) Copyright © 2013, 2009, 2006 Pearson Education, Inc.
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Objective #4: Example
4b.
Multiply: 6
xy
2 (10 6
xy
2 (10 6
xy
2 60
x
10
y
2 12
x
3) 6
xy
2 2
y
2 6 3)
xy
18
xy
2 2 3 60 12 18
xy
2 Copyright © 2013, 2009, 2006 Pearson Education, Inc.
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Objective #4: Example
4b.
Multiply: 6
xy
2 (10 6
xy
2 (10 6
xy
2 60
x
10
y
2 12
x
3) 6
xy
2 2
y
2 6 3)
xy
18
xy
2 2 3 60 12 18
xy
2 Copyright © 2013, 2009, 2006 Pearson Education, Inc.
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Objective #4: Example
4c.
Multiply: (7
x
(7
x
x
y
)
x
F O 21
x
2 25
xy
6
y
2
y
21
x
2 7
xy
18
xy
6
y
2
y
) I L
y
) Copyright © 2013, 2009, 2006 Pearson Education, Inc.
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Objective #4: Example
4c.
Multiply: (7
x
(7
x
x
y
)
x
F O 21
x
2 25
xy
6
y
2
y
21
x
2 7
xy
18
xy
6
y
2
y
) I L
y
) Copyright © 2013, 2009, 2006 Pearson Education, Inc.
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Objective #4: Example
4d.
Multiply: (6
xy
2 ( ) (6
xy
2
xy
2
xy
2
A
2 (6
xy
36
B
2 25
x
2 2 Copyright © 2013, 2009, 2006 Pearson Education, Inc.
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Objective #4: Example
4d.
Multiply: (6
xy
2 ( ) (6
xy
2
xy
2
xy
2
A
2 (6
xy
36
B
2 25
x
2 2 Copyright © 2013, 2009, 2006 Pearson Education, Inc.
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