3.2 The Remainder Theorem.

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Presentation transcript:

3.2 The Remainder Theorem

Homework from last day P. 124 #1 – 5 And from Tuesday, p. P. 114 #1, 2, 3, 5, 6, 9, C4

The Remainder Theorem Given P(x) = x3 - 4x2 + 5x + 1, determine the remainder when P(x) is divided by x - 1. -1 1 -4 5 1 The remainder is 3. -1 -2 3 1 -3 2 3 NOTE: P(1) gives the same answer as the remainder using synthetic division. Using f(x) = x3 - 4x2 + 5x + 1, determine P(1): P(1) = (1)3 - 4(1)2 + 5(1) + 1 = 1 - 4 + 5 + 1 = 3 Therefore P(1) is equal to the remainder. In other words, when the polynomial x3 - 4x2 + 5x + 1 is divided by x - 1, the remainder is P (1).

Remainder Theorem: When a polynomial P(x) is divided by x - a, the remainder is P(a). [think x - a, then x = a] Determine the remainder when x3 - 4x2 + 5x - 1 is divided by: a) x - 2 b) x + 1 Calculate P(-1) P(-1) = (-1)3 - 4(-1)2 + 5(-1) - 1 = -1 - 4 - 5 - 1 = -11 Calculate P(2) P(2) = (2)3 - 4(2)2 + 5(2) - 1 = 8 - 16 + 10 - 1 = 1 The remainder is -11. The remainder is 1. Point (-1, -11) is on the graph of of f(x) = x3 - 4x2 + 5x - 1 Point (2, 1) is on the graph of of f(x) = x3 - 4x2 + 5x - 1

Applications When the remainder is 30. is divided by Determine the value of k.

Problem Solving When the polynomial 3x3 + ax2 + bx -9 is divided by x - 2 , the remainder is -5. When the polynomial is divided by x + 1, the remainder is -16. What are the values of a and b?

Assignment Page 124 6a,7b, 8a,c, 9, 11, 14

Using Synthetic Division 1. (4x3 - 11x2 + 8x + 6) ÷ (x - 2) -2 4 -11 8 6 P(x) = (x - 2)(4x2 - 3x + 2) + 10 - 8 6 -4 4 -3 2 10 2. (2x3 - 2x2 + 3x + 3) ÷ (x - 1) -1 2 -2 3 3 -2 0 -3 P(x) = (x - 1)(2x2 + 3) + 6 2 0 3 6