9.4 Polynomial Division, Factors, and Remainders ©2001 by R. Villar All Rights Reserved

Polynomial Division, Factors, and Remainders In this section, we will look at two methods to divide polynomials: long division (similar to arithmetic long division) synthetic division (a quicker, short-hand method) Let’s take a look at long division of polynomials...

Example: Divide (2x 2 + 3x – 4) ÷ (x – 2) (x – 2) 2x 2 + 3x – 4 Rewrite in long division form... divisor dividend Think, how many times does x go into 2x 2 ? 2x Multiply by the divisor. 2x 2 – 4x Subtract. 7x – 4 Think, how many times does x go into 7x ? + 7 7x – remainder 2x x – 2 divisor Write the result like this...

Example: Divide (p 3 – 6) ÷ (p – 1) (p – 1) p 3 + 0p 2 + 0p – 6 Be sure to add “place-holders” for missing terms... p2p2 p 3 – p 2 p 2 + 0p + p p 2 – p p – 6 p 2 + p + 1 – 5 p – p – 1 –5 Let’s look at an abbreviated form of long division, called synthetic division...

Synthetic division can be used when the divisor is in the form (x – k). Example: Use synthetic division for the following (2x 3 – 7x 2 – 8x + 16) ÷ (x – 4) First, write down the coefficients in descending order, and k of the divisor in the form x – k : k 2 Bring down the first coefficient. 8 Multiply this by k 1 Add the column. 4 –4 –16 0 These are the coefficients of the quotient (and the remainder) 2x 2 + x – 4 Repeat the process

Example: Divide (5x 3 + x 2 – 7) ÷ (x + 1) – –7 Notice that k is –1 since synthetic division works for divisors in the form (x – k). place-holder 5x 2 – 4x + 4 – 11 x –5 –4 4 4 –11