2. Common Factor and Difference of Squares
Factoring means writing an expression as the product of its simplest factors.
Example 1: Factoring a number
14 = 7 × 2
[7 and 2 are the simplest factors of 14. We can't break it down any more than this.]
Example 2: Factoring an algebraic expression
3x + 15 = 3(x + 5)
This means that the factors of 3x + 15 are
3, and
(x + 5)
To be able to factor successfully, we need to recognise the formulas from Section 1. So it's a good idea to learn those formulas well!
Factoring Difference of Two Squares
To factor the difference of 2 squares, we just apply the formula given in Section 1 - Special Products in reverse. That is:
x2 − y2 = (x + y)(x − y)
Example 3: Factoring difference of 2 squares
Factor 36s2 − 121t2
Answer
We first recognize it is a difference of 2 squares, then we use the formula given above.
36s2 − 121t2
= (6s)2 − (11t)2
= (6s − 11t)(6s + 11t)
Exercises
Factor the following:
(1) 18p3 − 3p2
Answer
We note that 3p2 divides evenly into the 2 terms in the question. So the factorization is given by:
18p3 − 3p2 = 3p2(6p − 1)
(2) 5a + 10ax − 5ay + 20az
Answer
5a + 10ax − 5ay + 20az = 5a(1 + 2x − y + 4z)
(3) 36a2b 2 − 169c2
Answer
This is a difference of 2 squares.
36a2b2 − 169c2
= (6ab)2 − (13c)2
= (6ab + 13c)(6ab − 13c)
(4) (a − b)2 − 1
Answer
Once again, we recognize this as a difference of 2 squares.
(a − b)2 − 1 = (a − b)2 − 12
Put X = a − b and Y = 1
So (a − b)2 − 1
= X2 − Y2
= (X + Y)(X − Y)
= (a − b + 1)(a − b − 1)
(5) y4 − 81
Answer
y4 − 81
= (y2)2 − (9)2
= (y2 + 9)(y2 − 9)
= (y2 + 9)(y + 3)(y − 3)
(6) r2 − s2 + 2st − t2
Answer
We recognize that this involves 2 differences of two squares. We group it as follows:
r2 − s2 + 2st − t2
= r2 − (s2 − 2st + t2)
We recognize that s2 − 2st + t2 is a square, and equals (s − t)2. So we can factor our expression as follows:
r2 − s2 + 2st − t2
= r2 − (s2 − 2st + t2)
= r2 − (s − t)2
[This is also a difference of 2 squares.]
= [r − (s − t)][r + (s − t)]
= (r − s + t)(r + s − t)
