Section 6.3

Special Factoring

Overview

• In this section we discuss factoring of special polynomials.

• Special polynomials have a certain number of terms.

• The terms have special characteristics.• As a result, these special polynomials can be

factored by using patterns.

The Difference of Squares

• a2 – b2 = (a + b)(a – b)• The first term is a perfect square• The last term is a perfect square• There is a subtraction sign between the terms• 1, 4, 9, 16, 25, 36, 49, 64, 81, 100,…• Variables with even exponents• The sum of squares, a2 + b2, does not factor!

Examples

• k2 – 9• 36m2 – 25• 32c2 – 98d2

• (h + k)2 – 9

The Sum and Difference of Cubes

• a3 + b3 = (a + b)(a2 – ab + b2)• a3 – b3 = (a – b)(a2 + ab + b2)• The first term is a perfect cube• The last term is a perfect cube• 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000,…• Variables with exponents that are multiples of

3• The trinomial in the pattern is prime!

Examples

• y3 – 64• r3 + 343• 64g3 – 27h3

• 250x3 + 16y3

Perfect Square Trinomials

• a2 + 2ab + b2 = (a + b)(a + b) = (a + b)2

• a2 – 2ab + b2 = (a – b)(a – b) = (a – b)2

• The first term is a perfect square• The last term is a perfect square• The last sign is positive• Warning: check it by FOIL!• Other trinomial factoring methods will work

on these as well.

Examples

• x2 + 10x + 25• 9y2 – 6yz +z2

• (a – b)2 + 8(a – b) + 16

Don’t Forget!!!

• Always first look for a GCF!!!!!