Factoring a difference of two squares

conjugate pairs, difference of two squares, quadratic form, factoring, product of monomials

Once students are familiar with the idea of factoring, there are shortcuts to the FOIL method of quadratic equation factoring.  One of these is factoring a difference of two squares.  A difference of two squares means that we have a monomial multiplied by another monomial to give us a quadratic function where one monomial is a conjugate pair of the other and the middle term in the quadratic form of an equation disappears.

A quadratic equation is represented by ax²+bx+c, when the "bx" term disappears, we have a difference of two squares.

Example:

(x-3)*(x+3), here (x+3) is a conjugate pair of (x-3), and here is where we get a difference of two squares.

(x-3)(x+3), using the FOIL  method, is:

(x)(x)+(x)(3)+(-3)(x)+(-3)(3)=
x²+3x-3x-9
x²-9.  

Here, we see that since the middle term disappeared, we see that (x-3)(x+3) is simply x²-3², the two squares being x and 3, which is why it is called the difference of two squares.

Of course, the same thing would result if we were doing (x+3)(x-3) since multiplication is commutative.

Now, to give more examples:

(x+4)(x-4)= (x²-4²)=(x²-16)

(x+9)(x-9)=(x²-9²)=(x²-81)

conjugate pairs, difference of two squares, quadratic form, factoring, product of monomials