Factoring Polynomials
A strategy for factoring polynomials is given in the following box.
Strategy for Factoring Polynomials
1. If there are any common factors, factor them out first.
2. When factoring a binomial, look for the special cases: difference of two
squares, difference of two cubes, and sum of two cubes. Remember that a
sum of two squares a^{2} + b^{2} is prime.
3. When factoring a trinomial, check to see whether it is a perfect square
trinomial.
4. When factoring a trinomial that is not a perfect square, use grouping or trial
and error.
5. When factoring a polynomial of high degree, use substitution to get a
polynomial of degree 2 or 3, or use trial and error.
6. If the polynomial has four terms, try factoring by grouping.
Example
Using the factoring strategy
Factor each polynomial completely.
a) 3w^{3}  3w^{2}  18w
b) 10x^{2} + 160
c) 16a^{2}b 80ab + 100b
d) aw + mw + az + mz
Solution
a) The greatest common factor (GCF) for the three terms is 3w:
3w^{3}  3w^{2}  18w 
= 3w(w^{2}  w  6) 
Factor out 3w. 

= 3w(w  3)(w + 2) 
Factor completely. 
b) The GCF in 10x^{2} + 160 is 10:
10x^{2} + 160 = 10(x^{2} + 16)
Because x^{2} + 16 is prime, the polynomial is factored completely.
c) The GCF in 16a^{2}b  80ab + 100b is 4b:
16a^{2}b  80ab + 100b 
= 4b(4a^{2}  20a + 25) 

= 4b(2a  5)^{2} 
d) The polynomial has four terms, and we can factor it by grouping:
aw + mw + az + mz 
= w(a + m) + z(a + m) 

= (w + z)(a + m) 
Helpful hint
When factoring integers, we
write 4 = 2 Â· 2. However,
when factoring polynomials
we usually do not factor any of
the integers that appear. So
we say that 4b(2a  5)^{2} is
factored completely.