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Section 1.5 : Factoring Polynomials

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

\[2x{\left( {{x^2} + 1} \right)^3} - 16{\left( {{x^2} + 1} \right)^5}\]

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The first step is to identify the greatest common factor. In this case it looks like we can factor a 2 and an \({\left( {{x^2} + 1} \right)^3}\) out of each term and so the greatest common factor is \(2{\left( {{x^2} + 1} \right)^3}\) .

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Okay, now let’s do the factoring.

\[2x{\left( {{x^2} + 1} \right)^3} - 16{\left( {{x^2} + 1} \right)^5} = \require{bbox} \bbox[2pt,border:1px solid black]{{2{{\left( {{x^2} + 1} \right)}^3}\left( {x - 8{{\left( {{x^2} + 1} \right)}^2}} \right)}}\]

Don’t get excited if the greatest common factor has more “complicated” terms in it as this one did. The greatest common factor won’t always be just variables to powers.