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Section 1.5 : Factoring Polynomials

18. Factor the following polynomial.

\[2{x^{14}} - 512{x^6}\]

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Don’t let the fact that this polynomial is not quadratic worry you. Just because it’s not a quadratic polynomial doesn’t mean that we can’t factor it.

For this polynomial note that we can factor a \(2{x^6}\) out of each term to get,

\[2{x^{14}} - 512{x^6} = 2{x^6}\left( {{x^8} - 256} \right)\] Show Step 2

Now, notice that the second factor is a difference of perfect squares and so we can further factor this as,

\[2{x^{14}} - 512{x^6} = 2{x^6}\left( {{x^4} + 16} \right)\left( {{x^4} - 16} \right)\] Show Step 3

Next, we can see that the third term is once again a difference of perfect squares and so can also be factored. After doing that the factoring of this polynomial is,

\[2{x^{14}} - 512{x^6} = 2{x^6}\left( {{x^4} + 16} \right)\left( {{x^2} + 4} \right)\left( {{x^2} - 4} \right)\] Show Step 4

Finally, we can see that we can do one more factoring on the last factor.

\[2{x^{14}} - 512{x^6} = \require{bbox} \bbox[2pt,border:1px solid black]{{2{x^6}\left( {{x^4} + 16} \right)\left( {{x^2} + 4} \right)\left( {x + 2} \right)\left( {x - 2} \right)}}\]

Do not get too excited about polynomials that have lots of factoring in them. They will happen on occasion so don’t worry about it when they do.