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# 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 to 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 consists of two variables or terms separated by either the 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:*

**F**irst, which means multiplying the first terms together, i.e. (**a**+ b) (**c**+ d)**O**uter means that we multiply the outermost terms when the binomials are placed side by side, i.e. (**a**+ b) (c +**d**).**I**nner means multiply the innermost terms together i.e. i.e. (a +**b**) (**c**+ d).**L**ast. 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 the 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 the 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 we can use it to manipulate numbers, regardless of how they might look ugly with fractions and negative signs.

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

To master the foil method better, we shall solve a few examples of binomials.

*Example 1*

Multiply (2*x* + 3) (3*x* – 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:(-7*x*−3) (−2*x*+8)

__Solution__

- Multiply the first term:

= -7x * -2x = 14x ^{2}

- Multiply the outer terms:

= -7x * 8 = -56x

- Multiply the inner terms of the binomial:

= – 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) = –9*x*

- Multiply the inner terms of the binomial:

= (–3) * (2*x*) = –6*x*

- 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)] [3*x* + (2*y* + 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) = 2y
^{2} - (
*y*) *(1) =*y* - (–4) * (2
*y*) = –8*y* - (–4) * (1) = –4

Sum up the totals and collect the like terms:

= 2y^{2} – 7y – 4

Now replace this answer into the two binomials:

= 3x ^{2} + 5xy – 11x + (y – 4) (2y + 1) = 3x ^{2} + 5xy – 11x + 2y^{2} – 7y – 4

Therefore,

[*x* + (*y* – 4)] [3*x* + (2*y* + 1)] = 3x ^{2} + 5xy – 11x + 2y^{2} – 7y – 4

*Practice Questions*

Multiply the following binomials using the foil method:

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

__Answers__

- x
^{2}– 1 - – 4x
^{2}+ x +5 - 6x
^{2 }-11x -7 - x
^{2 }+ 2x -15 - 2x
^{2}-23x – 12 - – 40x
^{2}+46x +42

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