Multiplying binomials
The most commonly method used when multiplying binomials is FOIL. But what is FOIL?
Carefully examine what each letter stands for.However, do not focus on remembering since I will show below that it is important to master multiplication of binomials.
The letter F stands for first
The letter O stands for Outer
The letter I stands for inner
The letter L stands for last
I will illustrate with an example.Let's say that you want to multiply (x + 2) and (x + 3)
(x + 2) * (x + 3)
F: Multiplying the first term of each binomial we get x * x = x
^{2}
O:Multiplying the outer term of each binomial, we get x * 3 = 3x
I:Multiplying the inner term of each binomial, we get 2 * x = 2x
L: Multiplying the last term of each binomial, we get 2 * 3 = 6
Combining everything, we get x
^{2} + 3x + 2x + 6 = x
^{2} + 5x + 6
Sounds easy, right?
Unfortunately, my 3 years of experience in teaching has shown me that using the acronym FOIL makes multipling binomials a lot harder.
Students tend to focus more on remembering positions of terms and their respective names as opposed to understanding a very simple technique
And it can be very confusing!
For instance, the position used for first is also used for outer and the one used for outer is also used for last.
The brain does need all that extravagance
Furthermore, when you get a problem like 2 * ( x + y) or (x + y) * (x + 2 + z), you wonder why FOIL was invented in the first place
When multiplying binomials, I suggest you forget about the abbreviation FOIL and just do the problem naturally as I am going to explain.
Look at the following illustration:
Think of the problem as multiplying the first term in the first binomial(binomial on the left) by each term in the second binomial(binomial on the right)
We get x
^{2} + 3x
Then, multiply the second term in the first binomial by each term in the second binomial
We get 2x + 6
Doing it this ensure that you are not thinking about inner, outer, or the like and get yourself confused
Other examples of multiplying binomials
1) (x + -2) * ( 1 + x)
First, do:
x * 1 = x and
x * x = x
^{2}
Then,do
-2 * 1 = -2
-2 * x = -2x
Putting it all together, we get:
x + x
^{2} + -2 + -2x = x
^{2} + -x + -2 ( x + -2x = -x)
2) (4x + 2) * (x + 1)
First, do:
4x * x = 4x
^{2} and
4x * 1 = 4x
Second, do:
2 * x = 2x and
2 * 1 = 2
Putting it all together, we get:
4x
^{2} + 4x + 2x + 2 = 4x
^{2} + 6x + 2
Beyond FOIL:
2 * (x + y) = 2x + 2y
(x + y) * (x + 2 + z)
First, do:
x * x = x
^{2}
x * 2 = 2x
x * z = xz
Then, do:
y * x = yx
y * 2 = 2y
y * z = yz
Putting it all together, we get:
x
^{2} + 2x + xz + yx + 2y + yz
Multiplying binomials should not be hard if you follow what I taught you here.