Section 3.3 Dividing Polynomials; Remainder and Factor Theorems

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Section 3.3Dividing PolynomialsRemainder 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
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Long Division of Polynomials with Missing Terms
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.
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Example
Divide using Long Division.
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• Dividing Polynomials Using
• Synthetic Division

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Comparison of Long Division and Synthetic
Division of X3 4x2-5x5 divided by x-3
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Steps of Synthetic Division dividing 5x36x8 by
x2
Put in a 0 for the missing term.
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Using synthetic division instead of long
division. Notice that the divisor has to be a
binomial of degree 1 with no coefficients.
Thus
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Example
Divide using synthetic division.
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• The Remainder Theorem

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If you are given the function f(x)x3- 4x25x3
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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Example
Use synthetic division and the remainder theorem
to find the indicated function value.
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• The Factor Theorem

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Solve the equation 2x3-3x2-11x60 given that 3
is a zero of f(x)2x3-3x2-11x6. 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 20 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)
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(a) (b) (c) (d)