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Factor higher polynomials by grouping terms

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Generally the first step of factoring anything is to factor out any common factors. Even if this doesn’t result in a complete or significant “piece” of the factoring process, the more that you can factor out at the beginning the smaller the numbers and terms will be, making all future factoring techniques easier to do.

The next most common way to factor after the method of coefficients is a polynomial version of this idea of “factoring out common terms”: factoring by grouping. This is when we factor by grouping terms we think are “similar” together and then factoring out any expressions common to all the terms of the individual group. The hope is that what remains becomes a common term between all the grouped terms. This is difficult to describe, but easier to see with an example.

The key idea here is that the grouping can be done with any number of terms in any combination, so long as what we have leftover is exactly the same in each group after factoring out the GCF from each group. But this means that the grouping needs to have groups of the same size. That is to say, you could group a function that has 9 terms as 3 groups of 3 terms, but not 1 group of 4 terms and 1 group of 5 terms.