Summary of factoring techniques

Here are the steps for factoring polynomials in Beginning 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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