Polynomials: The Remainder and Factor Theorems

1. = remainder = P(- 2) 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) = 2x3 – 8x2 + 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 = - 3 (b) Here c = - 2, so divide (synthetically) P(x) by x + 2. 2 - 8 0 45 - 4 24 - 48 - 2 | 2 - 12 24 - 3

2. Polynomials: The Remainder and Factor Theorems 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) = 2x3 – x2 + 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. Slide 2

3. Polynomials: The Remainder and Factor Theorems Try: For the polynomial, P(x) = x3 – x2 + 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. Slide 3

4. Polynomials: The Remainder and Factor Theorems END OF PRESENTATION Click to rerun the slideshow.