Technique for Factoring Polynomials
Factoring ax^2 + bx+c, Where a Does Not Equal 0, AC Method
- There are several steps to factoring the standard polynomial using the AC method. First, divide all terms by any common factors to reduce the maximum value size of the terms for easier mathematical manipulation. This process reduces the complexity of the numbers and lessens the chances of making mistakes.
The Product of a x c and a + c = b
- Next, multiply the coefficient a and constant c. Find two values that when multiplied together equal the value of a x c.
There will be many values that fit this criteria; however, the two values that are chosen must not only equal a x c when multiplied together, but also equal b when added together.
If there are no possible factor combinations of the product of a x c = b, then the polynomial is prime, and further factoring is not possible.
When the values sum add to b, replace the middle term in the polynomial with these two factors, followed by any previous variable. For example, the middle term could be split up in an equation this way: 2a^2 + 7x +5 = 2a^2 + 3x + 4x +5.
Greatest Common Factor
- Now that there are four terms to the polynomial value, eliminate any possible greatest common factor (GCF) for the first two values, and again for the second two values. The GCF should be written outside a set of parentheses while the remaining products should go inside the parentheses. For example, 2x( x - 1) is the form that each of the two groups within the polynomial value should take.
Check the answer by multiplying the values outside the parentheses with those inside the parentheses, just as you would any other distributive algebraic process. The final result of the multiplication should equal that of the original polynomial value.