Factoring Polynomial Guidelines
1. Always check for GCF first!
2. For binomials, after GCF, we have three options:
(a) Difference of Squares:
(a2 − b2 ) = (a + b)(a − b)
(b) Difference of Cubes:
(a3 − b3 ) = (a − b)(a2 + ab + b2 )
(c) Sum of Cubes:
(a3 + b3 ) = (a + b)(a2 − ab + b2 )
3. For trinomials, we get hints from the signs.
(a) Identify coefficients a, b, c and their signs.
i. If b and c are positive, both binomial factors will have a + in the middle.
ii. If b is negative and c is positive, both binomial factors will have a − in the
middle.
iii. If c is negative, one binomial factor will have a + and the other a −. The larger
factor has the same sign as b.
(b) Multiply a and c and look for factors of a · c that make b. If c is positive, we want
a sum of b. If c is negative, we want a difference of b.
i. If a = 1, the binomial factors will both have x as the first term and the factors
that make b as the second terms(with appropriate signs).
ii. If a 6= 1, then you have two options:
α) Factoring by Grouping: Keep the ax2 term and the c term and replace the
bx term with the factors of a · c that make it. Then factor by grouping(you
can use the box to organize this).
β) Bottom’s Up: Proceed as if a = 1. Set each binomial factor with x as
the first term. Plug the two factors of a · c that make b in as the second
terms. Divide each second term by a, simplifying where possible, and then
moving the denominator up in front of the x.
4. For four terms, we can use Factoring by Grouping. If grouping doesn’t work, we may
need to find a combination of some of the previous options.(e.g. Two terms form a
difference of squares and the other two form a difference of cubes.)
5. You can always check your work by multiplying out your factors using distribution or
the FOIL method.