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Sarah Byom and Samantha Kingery singaporeolevelmaths.com If a polynomial f(x) is divided by x-k, then the remainder is r=f(k) Example: Find the remainder when f(x)=3x2+7x-20 is divided by x-2. K=2 (plug 2 in for x), so r=f(2)=3(2)2+7(2)-20=12+14-20=6. The remainder is 6. Example: Find the remainder when f(x) = x3-x2+4x+10 is divided by x-3 Answer: Remainder = 40. F(x) has a factor x-k if and only if f(k)=0. Example: Is x-7 a factor of x2-10x+21? Test: (7)2-10(7)+21 = 0. Yes, the factors are (x-7) and (x-3). Use long division to find the quotient and remainder when 3x3+5x2+8x+7 is divided by 3x+2. 1x2+1x+2 3x+2 )3x2+5x2+8x+7 3x2+2x2 3x2+8x+7 3x2+2x 6x+7 6x+4 3 Quotient Dividend Multiply: 1x2(3x+2) Subtract Multiply: 1x(3x+2) Subtract Multiply: 2(3x+2) Remainder Example: Find the quotient and remainder when x3-x2+4x+10 is divided by x-3. Answer: x2+2x+10 R: 40 Synthetic division is the shortcut method for the division of a polynomial by a linear divisor x-k. Example: Divide 2x3-3x2-5x-12 by x-3. Zero of divisor 3 Line for products Line for sums 1. 2. 3. 2 -3 -5 -12 Dividend 6 9 12 2 3 4 0 Remainder Quotient Multiply the zero of divisor (3) by the first coefficient of the dividend (2). Write the product about the line and one column to the right. Add the next coefficient of the dividend to the product just found and record the sum below the line in the same column. Repeat the “multiply” and “add” steps until the last row is complete. Divide 3x3-2x2+x-5 by x-1 using synthetic division. Answer: 3x2+x+2 R: 3 Find all of the real zeros of f(x) =2x2-7x3-8x2+14x+8. Use the Rational Zeros Theorem: Factors of 8 : +1, +2, +4, +8 = +1, +2, +4, +8, +1/2 Factors of 2: +1, +2 Graph the function, and compare the x-intercepts with the list of zeros. Then use synthetic division with the possible zeros, which in this case would be 4 and -1/2. 4 2 -7 -8 14 8 -1/2 2 1 -4 -2 8 4 -16 -8 -1 0 2 2 1 -4 -2 0 2 0 -4 0 Factor (x-4)(x+1/2)(2x2-4) Write the function as a product of linear and irreducible quadratic factors f(x)=x3-x2+x-6 Answer: (x-2)(x2+x+3) Demana, Franklin D., Bert K. Waits, Gregory D. Foley, and Daniel Kennedy. Precalculus: Graphical, Numerical, Algebraic. 8th ed. Boston: Pearson Education, Inc. , 2011. Print