Summary of factoring techniques
Here are the steps for factoring polynomials in Intermediate 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 (§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
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.
(A non-constant polynomial
that is not the product of two non-constant polynomials
is called prime.)
- 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.
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