# Summary of factoring techniques

Here are the steps for factoring polynomials in Intermediate Algebra:
• If necessary, put the polynomial in standard form.
• If possible, pull out any factors common to all terms (§6.1).
• If there are four terms, try factoring by grouping (§6.1).
• If there are three terms, try factoring into two binomials (§§6.2&6.3) or factoring as a perfect square (§6.4).
• If there are two terms (or if you now have factors with two terms), try factoring as a sum or difference of squares or cubes (§6.4).
• Keep factoring the factors until you can factor no further (§6.5).
These techniques will work for all polynomials up to degree 2 and for some polynomials of higher degree.

For definiteness, here are the conditions that must be met for a polynomial (with rational coefficients) to be completely factored:

• The first factor must be a constant, except that (unless it is the only factor) we leave it out if it is 1 or use just a minus sign if it is −1.
• Every other factor must be a non-constant polynomial with integer coefficients and a positive leading coefficient.
• No factor's coefficients may have a common integer factor greater than 1.
• No factor may be a product of two non-constant polynomials.
The last of these is the one that can be hard to check and may require fancy techniques to fix. (A non-constant polynomial that is not the product of two non-constant polynomials is called prime.)
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