Greatest Common Factor
Expressions that are multiplied are called factors.
(x + 2)(x + 3) = x2 + 5x + 6
The 2 binomials on the left care called the factors of the trinomial on the right.
Writing a polynomial as the product of two or more other polynomials is called factoring.
If a polynomial cannot be factored, it is called prime.
When a polynomial is factored all the way to its prime factors, it is factored completely.
When we factor, we take a polynomial that is connected by addition or subtraction and convert it into an equivalent form where the expressions are connected by multiplication. In other words, we move it one rung up the order of operations! This is a good thing, because we can do things to factors that we cannot do to addends (cancel, for example).
The Greatest Common Factor (GCF) of a polynomial is the largest polynomial that is a factor of each of the terms of the polynomial. Pulling out the GCF should always be your first step in a factoring problem.
Factor: 2x2 + 4x + 8 = 2(x2 + 2x + 4)
Factor 3a4b6 + 9ab7 = 3ab6(a3 + 3b)
Pulling out the GCF is like backwards distributing.
Sometimes we have to factor out a negative number.
Factor: –8a + 16 = – 8( a – 2)
Factoring by Grouping
What is the GCF in the polynomial 2a(x + 3) – 5b(x + 3)?
It is (x + 3)! 2a(x + 3) – 5b(x + 3)
If we factor that out, we get (x + 3)(2a – 5b)
Factoring by grouping is frequently used if you have four terms:
x2 + 3x + 2x + 6 = x(x + 3) + 2( x + 3) = (x + 3)(x + 2)