Factoring Polynomials

When we factor polynomials the is to completely factor meaning we can not acheive any more factoring. The resultant equation or function will be composed of positive primes (prime numbers). If we wanted to factor 12 to a series of positive primes we could:

• $12 = 3 x 4$
• $= 3 x 2 x 2$

At the second bullet, the scalar (12) has been completely factored. We do the same thing with polynomials. Assume we had a polynomial: $a^{2} + 25$. We could factor this to be $(a + 5)(a + 5)$.

Greatest Common Factor

The first technique to try is to factor out the greatest common factor. For example, we see the relationship:

$a(b + c) = ab + ac$

On the right hand side of the equality, the greatest common factor is $a$. We reverse the distributive law, and voila, can factor to the left hand side. (Check out the additional links in the summary for a cheat sheet with a reminder of these rules. Here is an example:

Assume we are given: $2x^{2} - 18x - 72$. First, we can factor out the 2:

$2(x^{2} - 9x - 36)$

Then we can factor the expression in the paranthesis, yielding completely factored expression:

$2(x + 3)(x - 12)$

Supplemental Video