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Warm-up Factor each: 1. 27x 3 – 64 2. 2x 3 – 3x 2 + 4x – 6 (3x – 4)(9x 2 + 12x + 16)(2x 3 – 3x 2 ) + (4x – 6) x 2 (2x – 3) + 2(2x – 3) (2x – 3)(x 2 + 2)

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 – 14 10 remainder 2x + 7 + 10 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 – 1 + 1 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 : 4 2 –7 –8 16 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) –1 5 1 0 –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 + 1 5 –5 –4 4 4 –11

You Try: Divide (2x 4 + x 3 – 2x 2 + 9x + 5) ÷ (2x+ 1) (2x + 1) 2x 4 + x 3 – 2x 2 + 9x + 5 2x 4 + x 3 – 2x 2 + 9x x3x3 –2x 2 – x 10x + 5 x 3 – x + 5 – x+ 5 10x + 5 0

You Try: Divide (3x 4 + 12x 3 – 5x 2 – 18x + 8) ÷ (x + 4) –4 3 12 –5 –18 8 3x 3 – 5x – 2 3 –12 0 0 –5 20 2 –8 0

Assignment p. 484: 7 – 29 odd, skip 13