Dividing Polynomials

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6-3
Dividing Polynomials
Warm Up
Lesson Presentation
Lesson Quiz
Holt Algebra 2
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Warm Up Divide using long division.
1. 161 7
23
2. 12.18 2.1
5.8
Divide.
3.
2x 5y
4.
7a b
3
Objective
Use long division and synthetic division to
divide polynomials.
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Vocabulary
synthetic division
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Polynomial long division is a method for dividing
a polynomial by another polynomials of a lower
degree. It is very similar to dividing numbers.
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Example 1 Using Long Division to Divide a
Polynomial
Divide using long division.
(y2 2y3 25) (y 3)
Step 1 Write the dividend in standard form,
including terms with a coefficient of 0.
2y3 y2 0y 25
Step 2 Write division in the same way you would
when dividing numbers.
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Example 1 Continued
Step 3 Divide.
2y2
5y
15
Notice that y times 2y2 is 2y3. Write 2y2 above
2y3.
Multiply y 3 by 2y2. Then subtract. Bring down
the next term. Divide 5y2 by y.
(2y3 6y2)
5y2 0y
(5y2 15y)
Multiply y 3 by 5y. Then subtract. Bring down
the next term. Divide 15y by y.
15y 25
Multiply y 3 by 15. Then subtract.
(15y 45)
70
Find the remainder.
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Example 1 Continued
Step 4 Write the final answer.
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Check It Out! Example 1a
Divide using long division.
(15x2 8x 12) (3x 1)
Step 1 Write the dividend in standard form,
including terms with a coefficient of 0.
15x2 8x 12
Step 2 Write division in the same way you would
when dividing numbers.
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Check It Out! Example 1a Continued
Step 3 Divide.
5x
1
Notice that 3x times 5x is 15x2. Write 5x above
15x2.
Multiply 3x 1 by 5x. Then subtract. Bring down
the next term. Divide 3x by 3x.
(15x2 5x)
3x 12
(3x 1)
Multiply 3x 1 by 1. Then subtract.
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Find the remainder.
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Check It Out! Example 1a Continued
Step 4 Write the final answer.
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Check It Out! Example 1b
Divide using long division.
(x2 5x 28) (x 3)
Step 1 Write the dividend in standard form,
including terms with a coefficient of 0.
x2 5x 28
Step 2 Write division in the same way you would
when dividing numbers.
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Check It Out! Example 1b Continued
Step 3 Divide.
x
8
Notice that x times x is x2. Write x above x2.
Multiply x 3 by x. Then subtract. Bring down
the next term. Divide 8x by x.
(x2 3x)
8x 28
(8x 24)
Multiply x 3 by 8. Then subtract.
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Find the remainder.
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Check It Out! Example 1b Continued
Step 4 Write the final answer.
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Synthetic division is a shorthand method of
dividing a polynomial by a linear binomial by
using only the coefficients. For synthetic
division to work, the polynomial must be written
in standard form, using 0 and a coefficient for
any missing terms, and the divisor must be in the
form (x a).
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Example 2A Using Synthetic Division to Divide by
a Linear Binomial
Divide using synthetic division.
(3x2 9x 2) (x )
Step 1 Find a. Then write the coefficients and a
in the synthetic division format.
Write the coefficients of 3x2 9x 2.
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Example 2A Continued
Step 2 Bring down the first coefficient. Then
multiply and add for each column.
1
3
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Step 3 Write the quotient.
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Example 2A Continued
3x2 9x 2
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Example 2B Using Synthetic Division to Divide by
a Linear Binomial
Divide using synthetic division.
(3x4 x3 5x 1) (x 2)
Step 1 Find a.
a 2
For (x 2), a 2.
Step 2 Write the coefficients and a in the
synthetic division format.
Use 0 for the coefficient of x2.
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Example 2B Continued
Step 3 Bring down the first coefficient. Then
multiply and add for each column.
3 1 0 5 1
2
Draw a box around the remainder, 45.
6
46
28
14
3
45
23
14
7
Step 4 Write the quotient.
Write the remainder over the divisor.
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Check It Out! Example 2a
Divide using synthetic division.
(6x2 5x 6) (x 3)
Step 1 Find a.
For (x 3), a 3.
a 3
Step 2 Write the coefficients and a in the
synthetic division format.
Write the coefficients of 6x2 5x 6.
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Check It Out! Example 2a Continued
Step 3 Bring down the first coefficient. Then
multiply and add for each column.
6 5 6
3
Draw a box around the remainder, 63.
18
69
6
63
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Step 4 Write the quotient.
Write the remainder over the divisor.
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Check It Out! Example 2b
Divide using synthetic division.
(x2 3x 18) (x 6)
Step 1 Find a.
For (x 6), a 6.
a 6
Step 2 Write the coefficients and a in the
synthetic division format.
Write the coefficients of x2 3x 18.
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Check It Out! Example 2b Continued
Step 3 Bring down the first coefficient. Then
multiply and add for each column.
1 3 18
6
There is no remainder.
6
18
1
0
3
Step 4 Write the quotient.
x 3
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You can use synthetic division to evaluate
polynomials. This process is called synthetic
substitution. The process of synthetic
substitution is exactly the same as the process
of synthetic division, but the final answer is
interpreted differently, as described by the
Remainder Theorem.
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Example 3A Using Synthetic Substitution
Use synthetic substitution to evaluate the
polynomial for the given value.
P(x) 2x3 5x2 x 7 for x 2.
2 5 1 7
2
Write the coefficients of the dividend. Use a
2.
4
34
18
2
41
17
9
P(2) 41
Check Substitute 2 for x in P(x) 2x3 5x2 x
7.
P(2) 2(2)3 5(2)2 (2) 7
?
P(2) 41
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Example 3B Using Synthetic Substitution
Use synthetic substitution to evaluate the
polynomial for the given value.
P(x) 6x4 25x3 3x 5 for x .
6 25 0 3 5
2
2
3
9
6
7
6
9
27
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Check It Out! Example 3a
Use synthetic substitution to evaluate the
polynomial for the given value.
P(x) x3 3x2 4 for x 3.
1 3 0 4
3
Write the coefficients of the dividend. Use 0 for
the coefficient of x2 Use a 3.
3
0
0
1
4
0
0
P(3) 4
Check Substitute 3 for x in P(x) x3 3x2 4.
P(3) (3)3 3(3)2 4
?
P(3) 4
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Check It Out! Example 3b
Use synthetic substitution to evaluate the
polynomial for the given value.
P(x) 5x2 9x 3 for x .
5 9 3
1
2
5
5
10
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Example 4 Geometry Application
Write an expression that represents the area of
the top face of a rectangular prism when the
height is x 2 and the volume of the prism is x3
x2 6x.
Substitute.
Use synthetic division.
1 1 6 0
2
The area of the face of the rectangular prism can
be represented by A(x) x2 3x.
2
6
0
1
0
3
0
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Check It Out! Example 4
Write an expression for the length of a rectangle
with width y 9 and area y2 14y 45.
The area A is related to the width w and the
length l by the equation A l ? w.
Substitute.
1 14 45
9
Use synthetic division.
9
45
1
0
5
The length of the rectangle can be represented by
l(x) y 5.
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Lesson Quiz
1. Divide by using long division. (8x3 6x2
7) (x 2)
2. Divide by using synthetic division. (x3
3x 5) (x 2)
3. Use synthetic substitution to evaluate
P(x) x3 3x2 6 for x 5 and x 1.
194 4
4. Find an expression for the height of a
parallelogram whose area is represented by
2x3 x2 20x 3 and whose base is
represented by (x 3).
2x2 7x 1