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How to Factor Mathematical Expressions

You often need to factor expressions (break those expressions into their simpler components, or factors) for calculus. Factoring means “unmultiplying,” like rewriting 12 as

image0.png

You don’t run across problems like that in calculus, however. For calculus, you need to be able to factor algebraic expressions, like factoring 5xy + 10yz as 5y(x + 2z). Algebraic factoring always involves rewriting a sum or difference of terms as a product.

The first step in factoring any type of expression is to pull out — in other words, factor out — the greatest thing that all of the terms have in common — that’s the greatest common factor, or GCF.

image1.png

Make sure you always look for a GCF to pull out before trying other factoring techniques.

After you pull out the GCF (if there is one), the next thing to do depends on whether you’re trying to factor a binomial (that’s a polynomial with two terms) or a trinomial (a three-term polynomial).

If you’re working on a binomial, you should look for one of the following three patterns. The first pattern is huge, the next two are much less important.

  • Difference of squares: Knowing how to factor the difference of squares is critical:

    image2.png

    Keep in mind that a difference of squares can be factored, but a sum of squares cannot be factored. In other words,

    image3.png

    is prime — you can’t break it up.

  • Sum and difference of cubes. You might also want to memorize the factor rules for the sum and difference of cubes:

    image4.png

    and

    image5.png
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