Special Products of Binomials
Two binomials with the same two terms but opposite signs separating the terms are called conjugates of each other. Following are examples of conjugates:

Example 1
Find the product of the following conjugates.
- (3 x + 2)(3 x – 2)
- (–5 a – 4 b)(–5 a + 4 b)
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Notice that when conjugates are multiplied together, the answer is the difference of the squares of the terms in the original binomials.
The product of conjugates produces a special pattern referred to as a difference of squares. In general,
( x + y)( x – y) = x 2 – y 2
The squaring of a binomial also produces a special pattern.
Example 2
Simplify each of the following.
- (4 x + 3) 2
- (6 a – 7 b) 2
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First, notice that the answers are trinomials. Second, notice that there is a pattern in the terms:
- The first and last terms are the squares of the first and last terms of the binomial.
- The middle term is twice the product of the two terms in the binomial.
The pattern produced by squaring a binomial is referred to as a square trinomial. In general,
Example 3
Do the following special binomial products mentally.
- (3 x + 4 y) 2
- (6 x + 11)(6 x – 11)
- (3 x + 4 y) 2 = 9 x 2 + 24 xy + 16 y 2
- (6 x + 11)(6 x – 11) = 36 x 2 – 121