Section 6-9
Factoring Special Products
There are two types of polynomials which occur often, and which you should
learn to recognize and be able to factor quickly; perfect squares of
binomials, and differences of squares.
SQUARE OF A BINOMIAL
If the coefficients of a trinomial ax2 + bx + c satisfy
the equation
then the trinomial is the perfect square of the binomial
Example 1:
Factor, if possible.
x2 – 14x + 49
Here, a = 1, b= –14, and c = 49. We have:
So, the trinomial is a perfect square:
You can verify this using FOIL.
Example 2:
Factor, if possible.
9w4 + 12w2 + 4
Here, a = 9, b = 12, and c = 4. (We can treat w2 as x,
and not worry about the fourth power.)
So, the trinomial is a perfect square:
This can also be verified using FOIL.
DIFFERENCE OF SQUARES
If a polynomial can be written as a2– b2,
then it can be factored as a difference of squares:
Example:
Factor, if possible.
9p2 – 49q2
This is a difference of squares with a = 3p and b =
7q.
9p2 – 49q2 =
(3p – 7q)(3p + 7q)
Again, you should verify this using FOIL.
