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# Product of a Sum and a Difference

If we multiply the sum a + b and the difference a - b by using FOIL, we get

 (a + b)(a - b) = a2 - ab + ab - b2 = a2 - b2

The inner and outer products add up to zero, canceling each other out. So the product of a sum and a difference is the difference of two squares, as shown in the following rule.

Rule for the Product of a Sum and a Difference

(a + b)(a - b) = a2 - b2

Example 1

Finding the product of a sum and a difference

Find the products.

a) (x + 3)(x - 3)

b) (a3 + 8)(a3 - 8)

c) (3x2 - y3)(3x2 + y3)

Solution

a) (x + 3)(x - 3) = x2 - 9

b) (a3 + 8)(a3 - 8) = a6 - 64

c) (3x2 - y3)(3x2 + y3) = 9x4 - y6

Helpful hint

You can use (a + b)(a - b) = a2 - b2 to perform mental arithmetic tricks such as 59 Â· 61 = 3600 - 1 = 3599. What is 49 Â· 51? 28 Â· 32?

The square of a sum, the square of a difference, and the product of a sum and a difference are referred to as special products. Although the special products can be found by using the distributive property or FOIL, they occur so frequently in algebra that it is essential to learn the new rules. In the next example we use the special product rules to multiply two trinomials and to square a trinomial.

Example 2

Using special product rules to multiply trinomials

Find the products.

a) [(x + y) + 3][(x + y) - 3]

b) [(m - n) + 5]2

Solution

a) Use the rule (a + b)(a - b) = a2 - b2 with a = x + y and b = 3:

 [(x + y) + 3][(x + y) - 3] = (x + y)2 - 32 = x2 +2xy + y2 - 9

b) Use the rule (a + b)2 = a2 + 2ab + b2 with a = m - n and b = 5:

 [(m - n) + 5]2 = (m - n)2 + 2(m - n)5 + 52 = m2 - 2mn + n2 + 10m - 10n + 25

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