Section 1.5 : Factoring Polynomials
18. Factor the following polynomial.
\[2{x^{14}} - 512{x^6}\]Show All Steps Hide All Steps
Start SolutionDon’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 2Now, 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 3Next, 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 4Finally, 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.