 # Foil Method to Distribute Two Binomials – Explanation & Examples

## What is the Foil Method?

Many students will start thinking of a kitchen when they first hear a mention of the term foil.

Here, we are talking about the FOIL – a mathematical series of steps used multiply two binomials. Before we learn what the term foil entails, let’s take a quick review of what the word binomial is.

A binomial is simply an expression that consist of two variables or terms which are separated by either addition sign (+) or subtraction sign (-). Examples of binomial expressions are 2x + 4, 5x + 3, 4y – 6, – 7y – y etc.

## How to do Foil Method?

The foil method is a technique used for remembering the steps required to multiply two binomials in an organized manner.

The F-O-I- L acronym stands for first, outer, inner and last.

Let’s explain each of these terms with the help of bold letters:

• First, which means, multiplying the first terms together i.e. (a + b) (c + d)
• Outer means that, the we multiply the outermost terms when the binomials are place side by side i.e. (a + b) (c + d).
• Inner means multiply the innermost terms together i.e. i.e. (a + b) (c + d).
• Last. This implies that, we multiply together the last term in each binomial i.e. i.e. (a + b) (c + d).

### How do you distribute binomials using foil method?

Let us put this method into perspective by multiplying two binomials, (a + b) and (c + d).

To find multiply (a + b) * (c + d).

• Multiply the terms which appear in the first position of binomial. In this, case a and c are the terms and their product are;

(a *c) = ac

• Outer(O) is the next word after the word first(F), therefore, multiply the outermost or the last terms when the two binomials are written side by side. The outermost terms are b and d.

(b * d) = bd

• The term inner implies that we multiply two terms that are in middle when the binomials are written side-by-side;

(b * c) = bc

• The last implies that we find the product of the last terms in each binomial. The last terms are b and d, therefore, b * d = bd

Now we can sum up the partial products of the two binomials beginning from the first, outer, inner and then the last. Therefore, (a + b) * (c + d) = ac + ad + bc + bd.

The foil method is an effective technique because it can be used to manipulate numbers, irregardless of how they might look ugly with fractions and negative signs.

### How do you multiply binomials using foil method?

In order to master the foil method better, we shall solve a few examples of binomials.

Example 1

Multiply (2x + 3) (3x – 1)

Solution

• Begin, by multiplying together, the first terms of each binomial

= 2x * 3x = 6x 2

• Now multiply the outer terms.

= 2x * -1= -2x

• Now multiply the inner terms.

= (3) * (3x) = 9x

• Finally multiply the last team in each binomial together.

= (3) * (–1) = –3

• Sum up the partial products starting from the first to last product and collect the like terms;

= 6x 2 + (-2x) + 9x + (-3)

= 6x 2 + 7x – 3.

Example 2

Use the foil method to solve:(-7x−3) (−2x+8)

Solution

• Multiply the firstterms;

= -7x * -2x = 14x 2

• Multiply the outer terms;

= -7x * 8 = -56x

• Multiply the inner terms of the binomials;

= – 3 * -2x = 6x

• Finally, multiply the last terms;

= – 3 * 8 = -24

• Find the sum of the partial products and collect the like terms;

= 14x 2 + ( -56x) + 6x + (-24)

= 14x 2 – 56x – 24

Example 3

Multiply (x – 3) (2x – 9)

Solution

• Multiply the first terms together;

= (x) * (2x) = 2x 2

• Multiply the outermost terms of each binomial;

= (x) *(–9) = –9x

• Multiply the inner terms of the binomials;

= (–3) * (2x) = –6x

• Multiply the last terms of each binomial;

= (–3) * (–9) = 27

• Sum up the products following the foil order and collect the like terms;

= 2x 2 – 9x -6x + 27

= 2x 2 – 15x +27

Example 4

Multiply [x + (y – 4)] [3x + (2y + 1)]

Solution

• In this case, the operations are broken down into smaller units and the results combine;
• Begin by multiplying the first terms;

= (x) * 3x = 3x 2

• Multiply the outer terms of each binomial;

= (x) * (2y + 1) = 2xy + x

• Multiply the inner terms of each binomial;

= (y – 4) (3x) = 3xy – 12x

• Now finish by multiplying the last terms

= (y – 4) (2y + 1)

Since the last terms area gain two binomials; Sum up the products;

= 3x 2 + 2xy + x + 3xy – 12x +(y – 4) (2y + 1)

= 3x 2 + 5xy – 11x + (y – 4) (2y + 1)

Again, apply the foil method on (y – 4) (2y + 1).

• (y) * (2y) = 2y2
• (y) *(1) = y
• (–4) * (2y) = –8y
• (–4) * (1) = –4

Sum up the totals and collect the like terms;

= 2y2 – 7y – 4

Now replace this answer into the two binomials;

= 3x 2 + 5xy – 11x + (y – 4) (2y + 1) = 3x 2 + 5xy – 11x + 2y2 – 7y – 4

Therefore,

[x + (y – 4)] [3x + (2y + 1)] = 3x 2 + 5xy – 11x + 2y2 – 7y – 4

### Practice Questions

Multiply the following binomials using the foil method:

1. (- x−1) (−x+1).
2. (4x+5) (x+1)
3. (3x−7) (2x+1)
4. (x+5) (x−3)
5. (x−12) (2x+1).
6. (10x−6) (4x−7)