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2.3 Polynomial Division and Synthetic Division

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

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2.3 Polynomial Division and Synthetic Division

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  1. 2.3 Polynomial Division and Synthetic Division Ex. Long Division What times x equals 6x3? 6x2 - 7x + 2 Change the signs and add. - + 6x3 - 12x2 - 7x2 + 16x - 4 - + - 7x2 + 14x 2x - 4 2x - 4

  2. 6x3 – 19x2 + 16x – 4 = (x – 2)(6x2 – 7x + 2) + 0 f(x) = d(x)q(x) + r(x) DividendDivisorQuotient Remainder

  3. Divide x3 – 1 by x - 1 x2 + x + 1 x3 - x2 - + x2 + 0x - 1 x2 - x - + x - 1 x - 1

  4. Synthetic Division Use synthetic division to divide x4 – 10x2 – 2x + 4 by x + 3. First, write the coef’s.of the dividend. Put zeros in for missing terms. Bring down the 1, mult. then add diagonally. 1 0 -10 -2 4 -3 1 -3 -1 1 1 remainder quotient x3 - 3x2 - 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) = 3x3 + 8x2 + 5x - 7. 3 8 5 -7 -2 3 1 -9 2 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) = 2x4 + 7x3 – 4x2 – 27x - 18 2 7 -4 -27 -18 2 2 11 18 0 f(2) = 0 9 f(-3) = 0 -3 2 5 3 0 (x – 2)(x + 3)(2x2 + 5x + 3)

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