1.) A binomial is an algebraic expression composed of 2 terms like 2x + y.
The FOIL method
The acronym F O I L stands for
First by First, [Outer by Outer + Inner by Inner], Last by Last.
These are instructions for finding the product of two binomials.
by, means multiplied by.
Say we want to find the product ( x – 3)(2x + 1).
Multiply the First terms in each bracket: x times 2x = 2x².
Multiply Outer terms: (x times +1) and Inner terms (–3 times 2x), then add = – 5x.
Multiply the Last terms in each bracket: – 3 times +1 = – 3.
So, ( x – 3) (2x + 1) = 2x² – 5x – 3.
Do the 2nd step [(O × O) + (I × I)] as a single step for greatest efficiency.
Note: F × F refers to the First term in each bracket.
O × O and I × I refers to the Outer and Inner terms of the expression as a whole
L × L refers to the Last term in each bracket.
Using this technique, let's find the product of (3x + 2)(2x – 3)
F by F is (3x)(2x) = 6x²
(O × O) + (I × I) is (3x)(– 3) + (+2)(2x) = – 5x
L by L is (+2)(– 3) = – 6
So (3x + 2)(2x – 3) = 6x² – 5x – 6
2.) Multiplying a Binomial by Itself
What happens when we square a binomial (in other words, multiply it by itself) .. ?
(a+b)2 = (a+b)(a+b) = ... ?
The result:
(a+b)2 = a2 + 2ab + b2
This illustration shows why it works:
(a+b)(a+b) in squares
2. Subtract Times Subtract
And what happens when we square a binomial with a minus inside?
(a−b)2 = (a−b)(a−b) = ... ?
The result:
(a−b)2 = a2 − 2ab + b2
If you want to see why, then look at how the (a−b)2 square is equal to the big a2 square minus the other rectangles:
(a-b)(a-b) in squares
(a−b)2 = a2 − 2b(a−b) − b2
= a2 − 2ab + 2b2 − b2
= a2 − 2ab + b2
