Our special products of binomials come through by noticing how expressions are simplified in binomial multiplication. Binomial multiplication is often modeled through a process known as FOIL, which is an acronym that stands for First, Outside, Inside, Last. It describes an order to multiply terms that make up the two factors in binomial multiplication.
Below is an example of using FOIL to multiply two binomials:
Square of a Binomial Sum
Take a look at what happens when we expand a binomial sum that is being squared:
Note the relationship between the constant in our binomial sum (3), the coefficient of the x-term (6), and the constant term in the expanded trinomial (9). See if the relationship you spotted holds true in another example:
What is the relationship between 5, 10, and 25? If we take the constant from the binomial sum, 5, and double this number, we get the coefficient of x-term, 10. And if we square 5, we get 25, which is the constant term in the expanded trinomial. This holds true for all squares of binomial sums.
Square of a Binomial Difference
Next, let's take a look at similar expressions, except these examples involve subtraction instead of addition:
Note that the only distinction here is that 6x is –6x, because we subtracted 3 from x in the binomial. We still have +9 because –3 times –3 is a positive number. Let's take a look at another example:
We can generalize this as the square of a binomial difference:
The expanded form for the square of a binomial sum and the square of a binomial difference are also referred to as perfect square trinomials, because they consist of three terms, which can be simplified into an expression squared.
Perfect Square Trinomial: a polynomial with three terms, we can be simplified as a binomial squared, or
Difference of Squares
So far we have discussed special products involving a binomial squared, where a is either positive or negative (which makes the distinction between addition/binomial sum, or subtraction/binomial difference. Now, we are going to consider the case where we have a binomial sum multiplied by a binomial difference.
There are a couple of things to note in this example:
We can generalize this as a difference of squares.
Difference of Squares: Two squared terms separated by subtraction, , which can be expressed as
two squared terms separated by subtraction, x^2 - a^2, which can be expressed as (x+a)•(x-a)
a polynomial with three terms, which can be simplified as a binomial squared, (x+a)^2