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Section 3.3 Dividing Polynomials; Remainder and Factor Theorems Long Division of Polynomials and The Division Algorithm Long Division of Polynomials Long Division of Polynomials 3x 4 3x 2 9 x 6 x 5 2 9x 6x 12 x 5 2 13 3x 2 12 x 8 13 Long Division of Polynomials with Missing Terms x5 2 x +5x -3 x 31x 17 2 x 5x 3 3x 2 3 x 3 +5x 2 3x -5x 6x 2 2 -5x 2 25x 15 31x- 17 You need to leave a hole when you have missing terms. This technique will help you line up like terms. See the dividend above. Example Divide using Long Division. 2 x 5 6 x3 4 x 2 +7 Example Divide using Long Division. x 2 2 x 1 8 x 4 3x3 +5x 1 Dividing Polynomials Using Synthetic Division Comparison of Long Division and Synthetic Division of X3 +4x2-5x+5 divided by x-3 Steps of Synthetic Division dividing 5x3+6x+8 by x+2 Put in a 0 for the missing term. Using synthetic division instead of long division. Notice that the divisor has to be a binomial of degree 1 with no coefficients. 2 5 + 7 - 1 10 6 5 3 5 Thus: 5 5x 3 x2 x 2 5x2 7 x 1 Example Divide using synthetic division. 3x3 5 x 2 7 x 8 x4 The Remainder Theorem If you are given the function f(x)=x3- 4x2+5x+3 and you want to find f(2), then the remainder of this function when divided by x-2 will give you f(2) f(2)=5 f (1) for f(x)=6x 2 2 x 5 is 1 6 -2 5 6 4 6 f(1)=9 4 9 Example Use synthetic division and the remainder theorem to find the indicated function value. f ( x) 3x3 5x2 1; f(2) The Factor Theorem Solve the equation 2x3-3x2-11x+6=0 given that 3 is a zero of f(x)=2x3-3x2-11x+6. The factor theorem tells us that x-3 is a factor of f(x). So we will use both synthetic division and long division to show this and to find another factor. Another factor Example Solve the equation 5x2 + 9x – 2=0 given that -2 is a zero of f(x)= 5x2 + 9x - 2 Example Solve the equation x3- 5x2 + 9x - 45 = 0 given that 5 is a zero of f(x)= x3- 5x2 + 9x – 45. Consider all complex number solutions. Divide x 3 x 2 x 8 x 3 2 2 x x 8 (a) 2 x 4x 2 (b) 2 x 4 x 14 (c) 34 (d) x 4 x 14 x3 2 Use Synthetic Division and the Remainder Theorem to find the value of f(2) for the function f(x)=x 3 +x 2 - 11x+10 (a) 2 (b) 0 (c) 5 (d) 12