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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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Dividing Polynomials Section 2.4. Objectives Divide two polynomials using either long division or synthetic division. Use the Factor Theorem to show that.

Dividing Polynomials Section 2.4. Objectives Divide two polynomials using either long division or synthetic division. Use the Factor Theorem to show that.

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