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Table of Contents Polynomials: The Remainder and Factor Theorems The remainder theorem states that if a polynomial, P(x), is divided by x – c, then the remainder equals P(c). Example 1:For the polynomial, P(x) = 2x 3 – 8x 2 + 45, (a)find P(- 2) by direct evaluation, (b)find P(- 2) using the remainder theorem. (a)P(- 2) = 2(- 2) 3 – 8(- 2) 2 + 45 (b)Here c = - 2, so divide (synthetically) P(x) by x + 2. 2- 80 45 - 4 24 - 48 - 2 | 2 - 12 24 - 3 = - 3 = remainder = P(- 2)

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Table of Contents Polynomials: The Remainder and Factor Theorems Slide 2 The factor theorem states that for a polynomial, P(x), if x – c is a factor, then P(c) = 0. Also, if P(c) = 0, then x – c is a factor. Example 2:For the polynomial, P(x) = 2x 3 – x 2 + 3x – 4, use the factor theorem to show that x – 1 is a factor. P(1) = 2(1) 3 – (1) 2 + 3(1) – 4= 0. Since P(1) = 0, x – 1 is a factor.

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Table of Contents Polynomials: The Remainder and Factor Theorems Slide 3 Try:For the polynomial, P(x) = x 3 – x 2 + x – 6, (a) find P(5) using the remainder theorem, (b)use the factor theorem to show that x – 2 is a factor. (a) 1- 1 1- 6 5 20 105 5 | 1 4 21 99 = P(5) (b)P(2) = (2) 3 – (2) 2 + (2) – 6 = 0 Since P(2) = 0, x – 2 is a factor.

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Table of Contents Polynomials: The Remainder and Factor Theorems

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Table of Contents Polynomials: Multiplicity of a Zero The polynomial, P(x) = x 3 – x 2 – 21x + 45, factors as, P(x) = (x – 3)(x – 3)(x + 5). Its zeros.

Table of Contents Polynomials: Multiplicity of a Zero The polynomial, P(x) = x 3 – x 2 – 21x + 45, factors as, P(x) = (x – 3)(x – 3)(x + 5). Its zeros.

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