Transcript Ch. 5.7 power point

Chapter 5
Section 7
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5.7
1
2
Dividing Polynomials
Divide a polynomial by a monomial.
Divide a polynomial by a polynomial.
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Objective 1
Divide a polynomial by a
monomial.
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Slide 5.7 - 3
Dividing a polynomial by a monomial.
We add two fractions with a common denominator as follows.
a b ab
 
c c
c
In reverse this statement gives a rule for dividing a polynomial
by a monomial:
To divide a polynomial by a monomial, divide each term of the
polynomial by the monomial:
ab a b
   c  0 .
c
c c
Dividend
25 2 5
Examples:
 
3
3 3
and
x  3z
x 3z


2y
2y 2y
Quotient
Divisor
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Slide 5.7 - 4
EXAMPLE 1
Dividing a Polynomial by a
Monomial
Divide 12m6 + 18m5 + 30m4 by 6m2.
Solution:
12m6  18m5  30m4
2
6m
6
5
4
12m 18m 30m



2
2
6m
6m
6m 2
 2m  3m  5m
4
3
2
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Slide 5.7 - 5
EXAMPLE 2
Divide
Dividing a Polynomial by a
Monomial
50m4  30m3  20m
.
3
10m
Solution:
50m4 30m3 20m



3
3
3
10m 10m 10m
 5m  3  2m
2
 5m  3  2
m
2
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Slide 5.7 - 6
EXAMPLE 3
Dividing a Polynomial by a
Monomial with a Negative
Coefficient
Divide −8p4 − 6p3 − 12p5 by −3p3.
Solution:
12 p5  8 p 4  6 p3

3 p3
12 p5 8 p 4 6 p3



3
3
3 p
3 p 3 p3
8p
 4p 
2
3
2
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Slide 5.7 - 7
EXAMPLE 4
Divide
Dividing a Polynomial by a
Monomial
45 x 4 y 3  30 x3 y 2  60 x 2 y
.
2
15 x y
Solution:
45x 4 y 3 30 x3 y 2 60 x 2 y



2
2
15x y 15 x y
15 x 2 y
 3x2 y 2  2 xy  4
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Slide 5.7 - 8
Objective 2
Divide a polynomial by a
polynomial.
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Slide 5.7 - 9
Divide a polynomial by a polynomial.
To divide a polynomial by a polynomial (other than a monomial).
Both polynomials must first be written in descending powers.
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Slide 5.7 - 10
Divide a polynomial by a polynomial. (cont’d)
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Slide 5.7 - 11
EXAMPLE 5
Dividing a Polynomial by a
Polynomial
2 x3  5 x  x 2  13
.
Divide
2x  3
1
x x 4 
2
x

3
3
2
Solution: 2 x  3 2 x  x  5 x  13
3
2
2 x  3x
2
2 x  5x
2
1
2 x  3x
x2  x  4 
8 x  13
2x  3
8 x  12
1
Remember to include “ 
2
remainder
” as part of the answer.
divisor
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Slide 5.7 - 12
EXAMPLE 6
Dividing into a Polynomial with
Missing Terms
Divide x3 − 8 by x − 2.
x 2x 4
x  2 x3  0 x 2  0 x  8
3
2
x  2x
2
2x  0x
2
2x  4x
4x  8
4x  8
0
2
Solution:
x2  2 x  4
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Slide 5.7 - 13
EXAMPLE 7
Dividing by a Polynomial with
Missing Terms
2m5  m4  6m3  3m2  18
Divide
.
2
m 3
3
2
6
2m  m
Solution:
m2  0m  3 2m5  m4  6m3  3m2  0m  18
2m5  0m4  6m3
m4  0m3  3m2
m4  0m3  3m2
6m2  0m  18
6m2  0m  18
0
2m3  m2  6
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Slide 5.7 - 14
EXAMPLE 8
Dividing a Polynomial when the
Quotient Has Fractional
Coefficients
Divide 3x3 + 7x2 + 7x + 11 by 3x + 6.
2 1 x 5
3
3
2
x
3x  6 3x3  7 x  7 x  11
3x3  6 x 2
x2  7 x
x2  2 x
5 x  11
5 x  10
1
5
1
2
x  x 
3
3 3x  6
1
Solution:
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
1
3x  6
Slide 5.7 - 15