FOILing and Explanation for FOIL 06:08 minutes

Video Transcript

Transcript FOILing and Explanation for FOIL

John Dolittler comes home from work and reads his mail. One letter catches his interest. It's a letter from a distant aunt. What's this? The letter says that he will inherit all of her money and all of her beloved animals from all over the world. John really loves animals, and has always wanted to have a little zoo. This is his big chance! Animals needs space, but that shouldn't be a problem with his new-found fortune. To help with his planning, John will use the FOIL method.

Multiplying two binomials

John finds an old map of his property and the nearby fields. But over time, the measurements of the fields have faded on the map. John knows that his property is 25 meters by 20 meters. He can't read the width of the field next to his property so he writes 'x' for its width. He also knows that the field above his property has a length that is twice as long as the other property's width, so it can be represented by 2x. If John buys both of these fields, the length of his new property would be 25 + 2x and the width would be 20 + x. For the total area, he multiplies these expressions by using the FOIL method. FOIL is an easy way to remember how to multiply two binomials.

The 'F' in FOIL stands for First, this means that you should multiply the two numbers that come first in each parentheses, 25 times 20 is 500. The 'O' stands for Outer, this means that you should multiply the two numbers on the outside. 25 times 'x' is 25x. The 'I' stands for Inner, the two inner numbers 2x times 20 make 40x. The 'L' stands for Last. You should multiply the numbers that come last in each set of parentheses. 2x times x is 2x squared.
After combining like terms and rearranging the terms in order of degree, we get 2x squared + 65x + 500. But FOIL only works when you are multiplying two binomials. So, why does it work anyway?

When we use the FOIL method, we are really just using the Distributive Property. The overall goal is to multiply every term in the first set of parentheses by every term in the second set of parentheses. We can do this by using the Distributive Property twice. Our first 'a' is the quantity 25 plus 2x, our first 'b' is 20 and our first 'c' is 'x'. Distributing the quantity 25 plus 2x over the quantity 20 plus 'x' gives us the following. Now, let's rearrange our equation to look like the definition of the distributive property.
What comes next? You guessed it! The distributive property comes to the rescue once again! 20 times 25 is 500. 20 times 2x is 40x. 'x' times 25 is 25x. 'x' times '2x' is 2x squared. After combining like terms and rearranging the terms by decreasing degree, we get 2x squared + 65x + 500. As you can see, it is the same as the FOIL method. But using FOIL cuts out a couple of steps.

Distributive Property

John isn't sure the area is big enough, so he's thinking about buying a third field. He knows that this field is three times as wide as the field next to his property. To calculate the bigger area, he rewrites the expression (25 + 2x) times (20 +x+3x). Since the FOIL method only works for multiplying two binomials, John has to use the Distributive Property. John breaks it up into three problems.
First, he multiplies (25 + 2x) by 20, then by x, and at last by 3x. By using the Distributive Property here, we get the following: 25 times 20 is 500, 2x times 20 is 40x, 25 times 'x' is 25x, 2x times 'x' is 2x squared, 25 times 3x equals 75x and 2x times 3x is 6x squared. After combining like terms, the area is 8x squared + 140x + 500. Don't forget to write the terms in order of decreasing degree. But careful, sometimes you can simplify first. Since 'x' and 3x are like terms, John could have combined them to get (20 + 4x) in the second parentheses.

Now we can use the FOIL method because it's the product of two binomials.

  • First: 25 times 20 is 500.
  • Outer: 25 times 4x is 100x.
  • Inner: 2x times 20 is 40x.
  • Last: 2x times 4x is 8x squared.

Look how big the new plot will be! John decides to buy the whole area for his little zoo.

John is just finishing up the zoo when the animals arrive. He's so excited that he can't wait to see what animals his aunt has collected from all over the world. What is this? Well, John tries to make the most out of his situation.