Factoring - Factoring Special Products
Objective: Identify and factor special products including a difference ofsquares, perfect squares, and sum
and difference of cubes.
When factoring there are a few special products that, if we can recognize them,can help us factor
polynomials. The first is one we have seen before. When multiplying special products we found that a
sum and a difference could multiply to a difference of squares. Here we will use this special product to
help us factor
Difference of Squares: a2 − b2 = (a + b)(a − b)
If we are subtracting two perfect squares then it will always factor to the sum and difference of the
x2 − 16 Subtracting two perfect squares,the square roots are x and 4
(x + 4)(x − 4) Our Solution
9a2 − 25b2Subtracting two perfect squares,the square roots are 3a and 5b
(3a +5b)(3a − 5b) Our Solution
It is important to note, that a sum of squares will never factor. It is alwaysprime. This can be seen if we
try to use the ac method to factor x2 + 36.
x2 + 36
No bx term, we use 0x.
x2 + 0x + 36
Multiply to 36, add to 0
1 · 36, 2 · 18, 3 · 12, 4 · 9, 6 · 6
No combinations that multiply to 36 add to 0Prime, cannot
factor Our Solution
It turns out that a sum of squares is always prime.
Sum of Squares: a2 + b2 = Prime