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 a2 + b2 is prime.
3. When factoring a trinomial, check to see whether it is a perfect square
4. When factoring a trinomial that is not a perfect square, use grouping or trial
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.
Using the factoring strategy
Factor each polynomial completely.
a) 3w3 - 3w2 - 18w
b) 10x2 + 160
c) 16a2b -80ab + 100b
d) aw + mw + az + mz
a) The greatest common factor (GCF) for the three terms is 3w:
|3w3 - 3w2 - 18w
||= 3w(w2 - w - 6)
||Factor out 3w.
||= 3w(w - 3)(w + 2)
b) The GCF in 10x2 + 160 is 10:
10x2 + 160 = 10(x2 + 16)
Because x2 + 16 is prime, the polynomial is factored completely.
c) The GCF in 16a2b - 80ab + 100b is 4b:
|16a2b - 80ab + 100b
||= 4b(4a2 - 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)
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