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Section 3.3Dividing Polynomials;
Remainder and Factor Theorems
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Long Division of Polynomials and
The Division Algorithm
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Long Division of Polynomials
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Long Division of Polynomials
2
3 2 9 6 5x x x
12 5x
4
13
3x
29 6x x
12 8x
13
3 2x
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Long Division of Polynomials with Missing Terms
2 3
3 2
2
2
x +5x -3 x 3x 2
x +5x 3x
-5x 6x 2
-5x 25x 15
31x- 17
5x 2
31 17
5 3
x
x x
You need to leave a hole when you have missing terms. This technique will help you line up like terms. See the dividend above.
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Example
Divide using Long Division.
3 22 5 6 4 +7x x x
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Example
Divide using Long Division.
2 4 32 1 8 3 +5 1x x x x x
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Dividing Polynomials Using
Synthetic Division
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Comparison of Long Division and Synthetic Division of X3 +4x2-5x+5 divided by x-3
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Steps of Synthetic Division dividing 5x3+6x+8 by x+2
Put in a 0 for the missing term.
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2 5 + 7 - 1
Using synthetic division instead of long division.
Notice that the divisor has to be a binomial of degree 1 with no coefficients.
5
103
65
2
55 3
22 5 7 1
xx
x x x
Thus:
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Example
Divide using synthetic division.
3 23 5 7 8
4
x x x
x
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The Remainder Theorem
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If you are given the function f(x)=x3- 4x2+5x+3 and you want to find f(2), then the remainder of this function when divided by x-2 will give you f(2)
f(2)=5
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2(1) for f(x)=6x 2 5 is
1 6 -2 5
6 4
6 4 9
f(1)=9
f x
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Example
Use synthetic division and the remainder theorem to find the indicated function value.
3 2( ) 3 5 1; f(2)f x x x
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The Factor Theorem
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Solve the equation 2x3-3x2-11x+6=0 given that 3 is a zero of f(x)=2x3-3x2-11x+6. The factor theorem tells us that x-3 is a factor of f(x). So we will use both synthetic division and long division to show this and to find another factor.
Another factor
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Example
Solve the equation 5x2 + 9x – 2=0 given that -2 is a zero of f(x)= 5x2 + 9x - 2
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Example
Solve the equation x3- 5x2 + 9x - 45 = 0 given that 5 is a zero of f(x)= x3- 5x2 + 9x – 45. Consider all complex number solutions.
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(a)
(b)
(c)
(d)
3 2Divide 2 8 3x x x x
2
2
2
2
8
4 2
4 14
344 14
3
x x
x x
x x
x xx
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(a)
(b)
(c)
(d)
3 2
Use Synthetic Division and the Remainder
Theorem to find the value of f(2) for the function
f(x)=x +x - 11x+10
2
0
5
12