Math: the conjugate, part 2: simplification

Tutoring math, polynomial structures arise. The tutor mentions using the conjugate.

The following is by my understanding.

Let’s imagine one is asked to simplify the following:

(x+3)(x-1)(x-3)(x+1)

If one does so in the order it’s presented, it gets a bit complicated. Using FOIL method on the first two brackets, then on the other two, one might arrive at

(x^2+2x-3)(x^2-2x-3)

Multiplying a trinomial by a trinomial isn’t that difficult, but a bit inconvenient.

Since the original question

(x+3)(x-1)(x-3)(x+1)

is multiplication, one can change its order:

(x+3)(x-3)(x-1)(x+1)

which, applying the idea of the conjugate (see my post from July 27, 2026), becomes

(x^2-9)(x^2-1)

Now, using FOIL, one can continue to simplify:

x^4 -x^2 – 9x^2 +9 = x^4 -10x^2 +9

One can see how the conjugate can ease simplification.

Source:

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

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.