Factoring Special Cases in Algebra

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Learn about factoring perfect square trinomials and the differences of two squares in algebra. Explore examples and understand how to factor expressions efficiently using these special cases.

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Warm Up 1.) What is the factored form of 2x2+ 9x + 10? 2.) What is the factored form of 6x2 23x 4?

8.7 Factoring Special Cases

Objective Factor perfect-square trinomials and the differences of two squares

Factoring Perfect Square Trinomials Reversing the rules for multiplying special case binomials will result in the original factors. Any trinomial of the form a2+ 2ab + b2or a2-2ab + b2is a perfect square trinomial because it is the result of squaring a binomial. For every real number a and b: a2+ 2ab + b2= (a + b)(a + b) = (a + b)2 x2+ 8x + 16 = (x + 4)(x + 4) = (x + 4)2 a2 2ab + b2= (a b)(a b) = (a b)2 4n2 12n + 9 = (2n 3)(2n 3) = (2n 3)2

Example 1 Factoring a Perfect-Square Trinomial What is the factored form of x2 4x +4?

Extra Example 1 What is the factored form of x2+ 6x + 9?

Example 2 Factoring to Find a Length Suppose the area of a square can be represented by the expression 9x2+ 24x + 16. What is an expression for the length of one side of the square?

Difference of Two Squares You can factor a difference of squares, a2 b2, as (a + b)(a b) For all real numbers a and b: a2 b2= (a + b)(a b) Ex.: x2 64 = (x + 8)(x 8) 25x2 36 = (5x + 6)(5x 6)