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# 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

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