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Section 1.5 : Factoring Polynomials

4. Factor out the greatest common factor from the following polynomial.

\[{x^2}\left( {2 - 6x} \right) + 4x\left( {4 - 12x} \right)\]

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The first step is to identify the greatest common factor and in this case we’ll need to be a little careful. If we just do a quick glance we might be tempted to just say the greatest common factor is just \(x\) since there is clearly an \(x\) in both terms.

However, notice that we can factor a 2 out of the \(4 - 12x\) in the second term to get,

\[{x^2}\left( {2 - 6x} \right) + 4x\left( {4 - 12x} \right) = {x^2}\left( {2 - 6x} \right) + 8x\left( {2 - 6x} \right)\]

Upon doing this we see that not only do we have an \(x\) in both terms we also have a \(2 - 6x\) in both terms and so the greatest common factor in this case is \(x\left( {2 - 6x} \right)\) .

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Okay, now let’s do the factoring.

\[{x^2}\left( {2 - 6x} \right) + 4x\left( {4 - 12x} \right) = {x^2}\left( {2 - 6x} \right) + 8x\left( {2 - 6x} \right) = \require{bbox} \bbox[2pt,border:1px solid black]{{x\left( {2 - 6x} \right)\left( {x + 8} \right)}}\]

Sometimes we need to do a little “pre factoring” work on a polynomial in order to determine just what the greatest common factor is. It won’t happen often, but it does need to be done often enough that we can’t forget about it.