synthetic division

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A method of finding the quotient and remainder when a polynomial f(x) is divided by a factor xh, in which the numbers are laid out in the form of a table. The rule to remember is that, at each step, ‘you multiply and then add’. To divide ax3+bx2+cx+d by xh, set up a table as follows:

Working from the left, each total, written below the line, is multiplied by h and entered above the line in the next column. The two numbers in that column are added to form the next total:

The last total is equal to the remainder, and the other numbers below the line give the coefficients of the quotient. By the Remainder Theorem, the remainder equals f(h).

For example, suppose that 2x3−7x2+5x+11 is to be divided by x−2. The resulting table is:

So the remainder equals 9, and the quotient is 2x2−3x−1. The calculations here correspond exactly to those that are made when the polynomial is evaluated for x=2 by nested multiplication, to obtain f(2)=9.

Subjects: Mathematics.

Reference entries