Synthetic DivisionDate: 12/22/96 at 23:58:18 From: David P Mcclain Subject: Synthetic division Please teach me about synthetic division! I am an over-the-hill college student and still don't understand it. Date: 12/24/96 at 15:02:24 From: Doctor Rob Subject: Re: Synthetic division Synthetic division is just a short-hand way of dividing a polynomial by a linear monic polynomial. In other words, one divides, using long division of polynomials, say: c3*x^2 + (c3*r+c2)*x + (c3*r^2+c2*r+c1) -------------------------------------------------- x - r ) c3*x^3 + c2*x^2 + c1*x + c0 c3*x^3 - c3*r*x^2 ----------------- (c3*r+c2)x^2 + c1*x (c3*r+c2)*x^2 - (c3*r^2+c2*r)*x ------------------------------- (c3*r^2+c2*r+c1)*x + c0 (c3*r^2+c2*r+c1)*x - (c3*r^3+c2*r^2+c1*r) ----------------------------------------- (c3*r^3+c2*r^2+c1*r+c0) The idea is to remove all the redundancy from this tableau. 1. By keeping columns of coefficients, we can eliminate all the powers of x, and the + signs between terms. c3 (c3*r+c2) (c3*r^2+c2*r+c1) -------------------------------------------------- 1 -r ) c3 c2 c1 c0 c3 -c3*r --------- c3*r+c2 c1 c3*r+c2 -(c3*r^2+c2*r) ----------------------- c3*r^2+c2*r+c1 + c0 c3*r^2+c2*r+c1 -(c3*r^3+c2*r^2+c1*r) ------------------------------------- c3*r^3+c2*r^2+c1*r+c0 2. The leading terms of the subtracted polynomials are always equal to the ones immediately above, so they are redundant and can be deleted. 3. The terms "brought down" are redundant and can thus be eliminated: c3 (c3*r+c2) (c3*r^2+c2*r+c1) -------------------------------------------------- 1 -r ) c3 c2 c1 c0 -c3*r --------- c3*r+c2 -(c3*r^2+c2*r) ----------------------- c3*r^2+c2*r+c1 -(c3*r^3+c2*r^2+c1*r) ------------------------------------- c3*r^3+c2*r^2+c1*r+c0 4. Instead of subtracting negatives, we can add positives. This means dropping all the - signs: c3 (c3*r+c2) (c3*r^2+c2*r+c1) -------------------------------------------------- 1 r ) c3 c2 c1 c0 c3*r ---- c3*r+c2 c3*r^2+c2*r -------------- c3*r^2+c2*r+c1 c3*r^3+c2*r^2+c1*r --------------------- c3*r^3+c2*r^2+c1*r+c0 5. The quotient above the line is the same as the quantities below the line (if we ignore the last one, which is the remainder), so it can be dropped. The horizontal line over the dividend can be removed, and all the other lines brought together to make a single horizontal line down below. The 1 is redundant, too, since it is always there, so it can be understood. r ) c3 c2 c1 c0 --- c3*r c3*r^2+c2*r c3*r^3+c2*r^2+c1*r --------------------------------------------------- c3 c3*r+c2 c3*r^2+c2*r+c1 | c3*r^3+c2*r^2+c1*r+c0 ----------------------- At this point, the mechanics of synthetic division can be readily seen: 1. Write r down, and put a line to the right and below it. This represents the divisor. 2. Further to the right, list the coefficients of the dividend polynomial. Don't forget to put in zeros if powers of the variable x are skipped. 3. Leave a blank line, then draw a horizontal line for the additions to follow. 4. Bring down the first coefficient below the line. 5. Multiply that last number by the number r, and put it above the line and in the next column to the right. 6. Add the two numbers in this column, and put the sum in the same column below the line. 7. If we are not out of columns, go back to step 5. 8. Draw a line to the left and below the last number in the last line. 9. The quotient is read off from the last line, coefficient by coefficient, left to right, ending at the vertical line. The remainder is the last number in the last line, to the right of the vertical line. Example: Divide (x - 3) into x^4 + x^3 - 7*x + 11 3 ) 1 1 0 -7 11 -- 3 12 36 87 ------------------- 1 4 12 29 | 98 ---- So the quotient is x^3 + 4*x^2 + 12*x + 29, and the remainder is 98. If the divisor is x + 7, then put r = -7, for instance. If the divisor is of the form 3*x - 14, then instead divide by x - 14/3, and then divide the quotient and remainder by 3, coefficient by coefficient. This obviously all works whatever the name of the variable (x in this message). Something more complicated must be used if the divisor has degree higher than 1. I hope that this is clear. If not, please write back for more help. -Doctor Rob, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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