Examples and practice with formula

What does "FOIL" stand for?

"Foil" is a way that people remember the 'formula' for multiplying binomials.

FOIL stands for first, outer, inner and last pairs . You are supposed to multiply these pairs as shown below!

Firsts: $$ x \cdot x= x^2 $$
Outer: $$ x \cdot 9 =9x $$
Inner: $$ 7 \cdot x =7x$$
Lasts: $$ 7 \cdot 9 = 63$$
FOIL explained binomials

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

Add up each pair!

$$ x^2 $$
9x
7x
+ 63
$$ x^2 + 16x + 63 $$

Video Multiplying Binomials

Visualizing with Rectangles and Area

What is the area of the rectangle below?

Foil as area of rectangles

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

Multiply the first, outer, inner and last pairs

Picture of Foil Method

First: x • x = $$ x^2 $$
Outer: x • 2 = 2x
Inner: 3 • x = 3x
Last: 3 • 2 = 6

Step 2

Simplify by adding the terms

$$ x^2 $$
2x
3x
+ 6
x2+ 5x + 6
Example 2

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Practice Problems

Problem 1

Multiply the 2 binomials below: $$(k + 7 )( k - 4) $$

Step 1

This is like example 1

Multiply the first, outer, inner and last pairs

Picture of Foil Method

First: k • k = k²
Outer: k • -4 = -4k
Inner: 7 • k = 7k
Last: 7 • -4 = -28

Step 2

Simplify by adding the terms

$$ k^2 $$
-4k
7k
+ (-28)
k2+ 3k - 28
Problem 2

Multiply the following binomials: $$(k - 3 )( k - 5) $$

Step 1

This is like example 1

Multiply the first, outer, inner and last pairs

Picture of Foil Method

First: k • k = k²
Outer: k • -5 = -5k
Inner: -3 • k = -3k
Last: -3 • -5 = 15

Step 2

Simplify by adding the terms

-5k
-3k
+ 15
k2 − 8k + 15
Problem 3

Multiply the binomials: $$ (2k + 9)(3k - 4) $$

Step 1

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

Picture of Foil Method

First: 2k • 3k = 6k²
Outer: 2k • -4 = -8k
Inner: 9 • 3 k = 27k
Last: 9 • -4 = -36

Step 2

Simplify by adding the terms

6k²
-8k
27k
+ (-36)
6k2+ 19k + -36
Problem 4

Multiply the 2 binomial :below : $$ (5k - 1 )(2k + 3) $$

Step 1

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

Picture of Foil Method

First: 5k • 2k = 10k²
Outer: 5k • 3 = 15k
Inner: -1 • 2k = -2k
Last: -1 • 3 = -3

Step 2

Simplify by adding the terms

10k²
15k
-2k
+ (-3)
  10k2+ 13k + -3
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