Summary of factoring techniques

• 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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