Factoring Special Polynomials(3.8)

Perfect Square Trinomials 4x x + 9 4x 2 + 6x + 6x + 9 (4x 2 + 6x) (+6x + 9) (2x + 3) (2x + 3) 2

A perfect square trinomial will be written in the form: a 2 + 2ab + b 2 or a 2 -2ab + b 2 When factoring a perfect square trinomial, the answer will be written in the form: (square root of a + square root of b) 2 or (square root of a – square root of b) 2 Looking at the example from the previous slide: 4x x + 9 4x 2 is a 2, 12x is 2ab, and 9 is b 2. For 12x, it breaks down to 2 (the square root of 4x 2 )(the square root of 9). This becomes 2(2x)(3). (2x)(3) = 6x 2(6x) = 12x. We can see that 4x x + 9 is in the form of a perfect square trinomial, therefore we can factor it as (2x + 3) 2.

Difference of Squares A difference of squares will always include the following: -An equation of the form a 2 – b 2 -The exponents in the equation will be even -The equation will always include subtraction -Any coefficients must be perfect squares

25x (5x + 6) (5x – 6) Example: All of the coefficients are perfect squares (25 and 36), the equation involves subtraction, and the exponent is even. Therefore this is a perfect square. When setting up the answer to factoring a difference of squares, there will be two sets of brackets. One bracket will be addition and the other will be subtraction. ( - ) ( + ) Then, square root both terms in the original equation and write the square roots in both of the new brackets. For this example, the answer would be:

Example 2: 16x 4 – y 4 Factoring the difference of squares, we get: (4x 2 + y 2 ) (4x 2 – y 2 ) With a problem like this, we will need to look closely at our second set of brackets. It is still a difference of squares. This means that we will need to factor it. Our answer will be: (4x 2 + y 2 ) (2x + y) (2x – y)