Before factoring any polynomial, write the polynomial indescending order of one of the variables. Then note how manyterms there are, and proceed by using one or more of thefollowing techniques. **1. **If there are** THREE TERMS**, look for these patterns:
**a. **Quadratic trinomials of the form ax+ bx + c where a = 1 (QT a = 1) factor into the product of twobinomials (double bubble) where the factors of c must add to b.
Example: x - 4x - 12 = (x - 6)(x + 2) **b. **Quadratic trinomials of the form ax+ bx + c where a 1 (QT a 1) eventually factor into the product of two binomials (doublebubble) but you must first find the factors of ac that add to b,rewrite the original replacing b with these factors of ac, thenfactor by grouping to finally get to the double bubble.
Example: 9x + 15x + 4 ac = (9)(4) = 36 factors of 36 that add to 15: 12 and 3 9x + 12x + 3x + 4 rewrite 15x as 12x + 3x, then factorby grouping 3x(x + 4) + 1(x + 4) = (x + 4)(3x + 1) **c.** Quadratic square trinomials (QST) of theform ax + bx + c may factor into the square of a binomial.Look for the pattern where two of the terms are perfect squares,and the remaining term is twice the product of the square root ofthe squares:
a ± 2ab + b= (a ± b) Example: 16x - 40x + 25 = (4x - 5) Note the pattern: Square root of 16 xis 4 x. Square root of 25 is 5. Twice the product of the square roots: 2(4x)(5) = 40x, whichis the middle term **2. **Factor all expressions completely.Sometimes, you will need to use two or three types of factoringin a single problem.
Example: -2x + 32 -2x + 32 = factor out the GCF of -2xand 32 -2(x - 16) = factor the difference of squares -2(x - 4)(x+ 4) = factor the remaining difference of squares -2(x - 2)(x + 2)(x+ 4) (remember that the sum of squares is prime) |