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# Using Patterns to Multiply Two Binomials

We obtain another useful pattern when we multiply the sum and difference of the same two terms.

 For example, letâ€™s use FOIL to find (a + b)(a - b). (a + b)(a - b) = Fa2 - Oab + Iba - Lb2 When we combine like terms, the middle terms cancel out. = a2 - b2

The expression a2 - b2 is called the difference of two squares because the operation is subtraction and each term is a square.

The pattern tells us that the product of conjugates always results in the difference of two perfect squares.

Formula â€” The Product of the Sum and Difference of the Same Two Terms

Let a and b represent any real numbers.

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

Note:

The following pairs of binomials are examples of conjugates:

a + b and a - b

x - y and x + y

w + 2 and w - 2

k - 5 and k + 5

Example

Find: (m + 8n)(m - 8n)

Solution

 The expression (m + 8n)(m - 8n) is in the form (a + b)(a - b). So we can use the shortcut: (a + b)(a - b) = a2 - b2 Substitute m for a and 8n for b.Simplify. So, (m + 8n)(m - 8n) = m2 - 64n2. (m + 8n)(m - 8n) = (m)2 - (8n)2= m2 - 64n2