FOIL - Multiplying Two Binomials

The heading may look a little strange. "FOIL' is an acronym for the way that you multiply two binomials together.

Multipling Two Binomials
Often when people see two binomials, such as this; (x+3)(2x-4), they will simplify it thus:

x(2x)+3(-4)=2x^2-12

However, this is largely incorrect. To multiply two binomials together, you must multiply every term in one binomial by every term in the other binomial.

There is an acronym that helps us remember how to do the multiplication correctly. That is; FOIL: First, Outside, Inner, Last. Below is a demonstration of how to do the multiplication correctly, using the FOIL method:

Simplify: (3x-1)(x+5)


 * First (3x-1)(x+5): 3x(x)=3x^2


 * Outside (3x-1)(x+5): 3x(5)=15x


 * Inner (3x-1)(x+5): -1(x) =-x

Now that you have all your terms figured out, you just put the terms together and combine like terms. 3x^2+15x-x-5 = 3x2+14x-5
 * Last (3x-1)(x+5): -1(5) =-5

It's as simple as that!

Proof of the FOIL Method
The FOIL Method makes perfect sense if you just use the distributive property - and a lot! First, here's a recap of the Distributive Property: a(b+c)=ab+ac. Alright, here goes:

Consider the first term to be equal to w, the second term to x... So what we have is: (w+x)(y+z)

Using the distributive property, we will call "a" the binomial "(w+x), "b' as "y", and "c" as "z". That will get us: (w+x)y+(w+x)z. Now we use the distributive property once more on each term, with "a" being "y" on the first term and "z" on the second. That finally gets us: wy+xy+wz+xz. Putting them in alphabetical order makes wy + wz + xy +xz. If you compare these to (w+x)(y+z), you will see:

First, Outer, Inner, Last

Practice Problems
This is just a little practice so that this information will lock in your memory. Plus its good so that you have some experience with this concept.

1. FOIL and simplify the expression (x+2)(x+1)

2. FOIL and simplify the expression (x-4)(2x+3)

3. FOIL and simplify the expression (2x-5)^2

4. FOIL and simplify the expression (7x-13)(5y+6)

5. FOIL and simplify the expression (4x-3)(4x+3)

6. FOIL and simplify the expression (4x^3-2y)(-2x^6+ 8x^5)

Answers will be found on the F Page.