- If necessary, put the polynomial in standard form.
- If possible, pull out any factors common to all terms (§7.1).
- If there are four terms, try factoring by grouping (§7.1).
- If there are three terms, try factoring into two binomials (§§7.2&7.3) or factoring as a perfect square (§7.4).
- 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 (§7.4).
- Keep factoring the factors until you can factor no further (§7.5).

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.

A product of two non-constant polynomials
is called a **composite** polynomial.
(The last rule above requires us to factor these polynomials further.)
A non-constant polynomial that is *not* composite
is called a **prime** polynomial.
(The constant polynomials are considered neither prime nor composite.)
Compare that a product of two whole numbers greater than 1
is called a *composite* number,
while a whole number greater than 1 that is *not* composite
is called a *prime* number.
(The whole numbers 0 and 1 are neither prime nor composite.
In this analogy,
the non-zero constant polynomials correspond to the whole number 1,
while the constant polynomial 0 corresponds to the whole number 0.)

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