The Square of a Binomial
To compute (a + b)2, the square of a binomial, we can write it as
(a + b)(a + b) and use FOIL:
| (a + b)2 |
= (a + b)(a + b) |
| |
= a2 + ab + ab + b2 |
| |
= a2 + 2ab + b2 |
So to square a + b, we square the first term (a2), add twice the
product of the two terms (2ab), then add the square of the last term (b2).
The square of a binomial occurs so frequently that it is helpful to learn this
new rule to find it. The rule for squaring a sum is given symbolically as
follows.
The Square of a Sum
(a + b)2 = a2 + 2ab + b2
Example 1
Using the rule for squaring a sum
Find the square of each sum.
a) (x + 3)2
b) (2a + 5)2
Solution
| a) (x + 3)2 = |
x2 |
+ 2(x)(3) |
+ 32 |
= x2 + 6x + 9 |
| |
↑ |
↑ |
↑ |
|
| |
Square of first |
Twice the procuct |
Square of last |
|
| b) (2a + 5)2 |
= (2a)2 + 2(2a)(5) + 52 |
| |
= 4a2 + 20a + 25 |
Caution
Do not forget the middle term when squaring a sum. The equation (x + 3)2
= x2 + 6x + 9 is an identity, but (x + 3)2 = x2
+ 9 is not an identity. For example, if x = 1 in (x + 3)2 = x2
+ 9, then we get 42 = 12 + 9, which is false.
When we use FOIL to find (a - b)2, we see that
| (a - b)2 |
= (a - b)(a - b) |
| |
= a2 - ab - ab + b2 |
| |
= a2 - 2ab + b2 |
So to square a - b, we square the first term (a2), subtract twice
the product of the two terms (-2ab), and add the square of the last term (b2).
The rule for squaring a difference is given symbolically as follows.
The Square of a Difference
(a - b)2 = a2 - 2ab + b2
Example 2
Using the rule for squaring a difference
Find the square of each difference.
a) (x - 4)2
b) (4b - 5y)2
Solution
| a) (x - 4)2 |
= x2 - 2(x)(4) + 42 |
| |
= x2 - 8x + 16 |
| b) (4b - 5y)2 |
= (4b)2 - 2(4b)(5y) + (5y)2 |
| |
= 16b2 - 40by + 25y2 |
|
