Raphael Stark

2022-12-26

How can you find the GCF of two polynomials?

juanitacelis1ovy

Expert

The greatest common factor of two polynomials and the greatest common factor of two integers share a similar idea.
The largest integer that is a factor of both the two integers is the greatest common factor of two integers. Finding the common primes and multiplying them together is one method of determining the GCF. Another method is to express the two integers as products of ascending primes.
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:
$P\left(x\right)=4{x}^{3}+24{x}^{2}+44x+24$
$Q\left(x\right)=6{x}^{3}+42{x}^{2}+84x+48$
To find the GCF of P(x) and Q(x) 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:
$GCF\left(P\left(x\right),Q\left(x\right)\right)=2\cdot \left(x+1\right)\left(x+2\right)=2\left({x}^{2}+3x+2\right)=2{x}^{2}+6x+4$

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