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:




Multiply first terms:  

Multiply outside terms:  

Multiply inside terms:  

Multiply last terms:  

Combine like terms 
Take a look at what happens when we expand a binomial sum that is being squared:

Square of a binomial sum  

Two factors of  

FOIL  

Combine like terms 
Note the relationship between the constant in our binomial sum (3), the coefficient of the xterm (6), and the constant term in the expanded trinomial (9). See if the relationship you spotted holds true in another example:

Square of a binomial sum  

Two factors of  

FOIL  

Combine like terms 
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 xterm, 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.
Next, let's take a look at similar expressions, except these examples involve subtraction instead of addition:

Square of a binomial sum  

Two factors of  

FOIL  

Combine like terms 
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:

Square of a binomial sum  

Two factors of  

FOIL  

Combine like terms 
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.
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.




FOIL  

Combine like terms 
There are a couple of things to note in this example:
We can generalize this as a difference of squares.