# How can you find the GCF of two polynomials?

May 31, 2015

The greatest common factor of a couple of polynomials is similar in concept to the greatest common factor of a couple of integers.

The greatest common factor of a couple of integers is the largest integer which is a factor of both of the integers. One way of finding the GCF is to express the two integers as products of ascending primes, identify the common primes and multiply them together.

The greatest common factor of a couple of polynomials is the largest polynomial which is a factor of both of the polynomials. To find the GCF of two polynomials we can factor both of them, identify the common factors and multiply them together.

For example, consider the two polynomials:

$P \left(x\right) = 4 {x}^{3} + 24 {x}^{2} + 44 x + 24$
$Q \left(x\right) = 6 {x}^{3} + 42 {x}^{2} + 84 x + 48$

To find the GCF of $P \left(x\right)$ and $Q \left(x\right)$ first factorize them:

$P \left(x\right) = 2 \cdot 2 \cdot \left(x + 1\right) \left(x + 2\right) \left(x + 3\right)$
$Q \left(x\right) = 2 \cdot 3 \cdot \left(x + 1\right) \left(x + 2\right) \left(x + 4\right)$

Picking out the common factors and multiplying them:

$G C F \left(P \left(x\right) , Q \left(x\right)\right) = 2 \cdot \left(x + 1\right) \left(x + 2\right) = 2 \left({x}^{2} + 3 x + 2\right) = 2 {x}^{2} + 6 x + 4$