Summary of factoring techniques
Here are the steps for factoring polynomials in Beginning Algebra:
These techniques will work
for all polynomials up to degree 2 and for some polynomials of higher degree.
- 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
For definiteness, here are the conditions that must be met
for a polynomial (with rational coefficients) to be completely factored:
The last of these is the one that
can be hard to check and may require fancy techniques to fix.
- 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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last edited on 2018 August 28.
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