## What "FOIL" stands for

"Foil" is a way that people remember the 'formula' for multiplying binomials. (An alternative approach is to use "double distributive" )

**FOIL ** stands for **f**irst, **o**uter, **i**nner and **l**ast pairs . You are supposed to multiply these pairs as shown below!

Firsts : | $$ x \cdot x= x^2 $$ |

Outers : | $$ x \cdot 9 =9x $$ |

Inners : | $$ 7 \cdot x =7x$$ |

Lasts : | $$ 7 \cdot 9 = 63$$ |

#### So, now that we've multiplied, what is next?

Add up each term!

$$ x^2 $$ |

9x |

7x |

+ 63 |

$$ x^2 + 16x + 63 $$ |

### Demonstration of FOIL

**Video** Multiplying Binomials

**Visualizing** with Rectangles and Area

#### What is the area of the rectangle below?

The area of each part of the rectangle can be seen as a 'part' of the FOIL formula

##### Example 1

Let's multiply the following binomials: (X + 3 )( X + 2)

Step 1
First: x • x = $$ x^2 $$

Outer: x • 2 = 2x

Inner: 3 • x = 3x

Last: 3 • 2 = 6

$$ x^2 $$ |

2x |

3x |

+ 6 |

x^{2}+ 5x + 6 |

##### Example 2

**Practice** Problems

This is like example 1

Multiply the first, outer, inner and last pairs

First: k • k = k²

Outer: k • -4 = -4k

Inner: 7 • k = 7k

Last: 7 • -4 = -28

Simplify by adding the terms

$$ k^2 $$ |

-4k |

7k |

+ (-28) |

k^{2}+ 3k - 28 |

This is like example 1

Multiply the first, outer, inner and last pairs

First: k • k = k²

Outer: k • -5 = -5k

Inner: -3 • k = -3k

Last: -3 • -5 = 15

Simplify by adding the terms

k² |

-5k |

-3k |

+ 15 |

k^{2} − 8k + 15 |

This is like example 1 with the slight twist that you now have to deal with coefficients in from of the variable of each binomial.

Multiply the first, outer, inner and last pairs

First: 2k • 3k = 6k²

Outer: 2k • -4 = -8k

Inner: 9 • 3 k = 27k

Last: 9 • -4 = -36

Simplify by adding the terms

6k² |

-8k |

27k |

+ (-36) |

6k^{2}+ 19k + -36 |

This is like example 1 with the slight twist that you now have to deal with coefficients in from of the variable of each binomial.

Multiply the first, outer, inner and last pairs

First: 5k • 2k = 10k²

Outer: 5k • 3 = 15k

Inner: -1 • 2k = -2k

Last: -1 • 3 = -3

Simplify by adding the terms

10k² | |

15k | |

-2k | |

+ (-3) | |

10k^{2}+ 13k + -3 |