1. 2.3 Polynomial Division and Synthetic Division Ex. Long Division What times x equals 6x 3 ? 6x 2 6x 3 - 12x 2 Change the signs and add. - + - 7x 2 + 16x - 4 - 7x - 7x 2 + 14x+ - 2x - 4 + 2 2x - 4

2. f(x) = d(x)q(x) + r(x) Dividend Divisor Quotient Remainder 6x 3 – 19x 2 + 16x – 4 = (x – 2) (6x 2 – 7x + 2) + 0

3. Divide x 3 – 1 by x - 1 x2x2 x 3 - x 2 x 2 + 0x - 1 + x x 2 - x x - 1 + 1 x - 1 - +

4. Synthetic Division Use synthetic division to divide x 4 – 10x 2 – 2x + 4 by x + 3. First, write the coef’s.of the dividend. Put zeros in for missing terms. 1 0-10-24 -3 Bring down the 1, mult. then add diagonally. 1-311 remainder quotient x 3 - 3x 2 - x + 1

5. Remainder Theorem: If a polynomial f(x) is divided by x – k, then the remainder is r = f(k) Use the remainder theorem to find f(-2) if f(x) = 3x 3 + 8x 2 + 5x - 7. 385-7 -23 2 1-9 f(-2) = -9 This means that (-2, -9) is a point on the graph of f.

6. Factor Theorem: A polynomial f(x) has a factor (x – k) if and only if f(k) = 0. Show that (x – 2) and (x + 3) are factors of f(x) = 2x 4 + 7x 3 – 4x 2 – 27x - 18 27-4-27-18 221118 9 0f(2) = 0 -32530 (x – 2)(x + 3)(2x 2 + 5x + 3) f(-3) = 0