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*)

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}

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