Algebra - Factoring Polynomials

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5. Factor the following polynomial by grouping.

\[7x + 7{x^3} + {x^4} + {x^6}\]

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Start Solution

The first step here is to group the first two term and the last two terms as follows.

\[\left( {7x + 7{x^3}} \right) + \left( {{x^4} + {x^6}} \right)\] Show Step 2

We can now see that we can factor a 7\(x\) out of the first grouping and an \({x^4}\) out of the second grouping. Doing this gives,

\[7x + 7{x^3} + {x^4} + {x^6} = 7x\left( {1 + {x^2}} \right) + {x^4}\left( {1 + {x^2}} \right)\] Show Step 3

Finally, we see that we can factor an \(x\left( {1 + {x^2}} \right)\) out of both of the new terms to get,

\[7x + 7{x^3} + {x^4} + {x^6} = \require{bbox} \bbox[2pt,border:1px solid black]{{x\left( {1 + {x^2}} \right)\left( {7 + {x^3}} \right)}}\]