Math: the conjugate

Tutoring math, the idea of a conjugate arises. The tutor mentions it.

The following is by my understanding.

In high school math, the conjugate refers to a binomial construction that, when multiplied by its own conjugate, yields no “middle terms.”

Example: What is the conjugate of x-5?

Solution: x+5 is the conjugate of x-5.

To see why x+5 is the conjugate of x-5, we multiply the two together using the FOIL method (see my post from November 26, 2012.)

(x-5)(x+5) = x^2-25

Note, in the product, the absence of an “x” term, which confirms that x+5 is the conjugate of x-5. Conversely, x-5 is the conjugate of x+5.

As a non-example, x-4 is not the conjugate of x-5, as follows:

(x-5)(x-4) = x^2-9x+20

In this product, the presence of the “x” term, -9x, aka the middle term, shows that x-4 isn’t the conjugate of x-5.

Conjugates are used for dividing and other purposes. Not only polynomials have them.

Source:

Travers, K.J., Dalton, L.C., Brunner, V.F., Taylor, A.R. (1977). Using Advanced Algebra. Doubleday Canada Limited.