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Section 2.4 Dividing Polynomials; Remainder and Factor Theorems

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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. (4x 2 – 8x + 6) ÷ (2x – 1)

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Example Divide using Long Division.

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Comparison of Long Division and Synthetic Division of X 3 +4x 2 -5x+5 divided by x-3

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Steps of Synthetic Division dividing 5x 3 +6x+8 by x+2 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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If you are given the function f(x)=x 3 - 4x 2 +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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Example Use synthetic division and the remainder theorem to find the indicated function value.

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Solve the equation 2x 3 -3x 2 -11x+6=0 given that 3 is a zero of f(x)=2x 3 -3x 2 -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 5x 2 + 9x – 2=0 given that -2 is a zero of f(x)= 5x 2 + 9x - 2

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Example Solve the equation x 3 - 5x 2 + 9x - 45 = 0 given that 5 is a zero of f(x)= x 3 - 5x 2 + 9x – 45. Consider all complex number solutions.

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