# Foil Method || Method to multiply binomials

In this post we will learn FOIL method of multiplying algebraic equation.

Foil method is a process to multiply two binomial expressions.

To understand the chapter you should have basic knowledge of algebraic expressions, constants and variables.

## What is FOIL method in Math?

The full form of FOIL is First, Outer, Inner and Last.

The method gives the instructions to multiply two binomial expression.

### What is Binomial Expression?

An algebraic expression containing two entities is called binomial expression. The entity can be constant, variable or both.

Examples of binomial expression;

\mathtt{\Longrightarrow \ 2\ +\ x}\\\ \\ \mathtt{\Longrightarrow \ x^{2} y\ +\ 9y}\\\ \\ \mathtt{\Longrightarrow \ 2x^{3} +\ 3y}

### Using FOIL to multiply binomial expression

Consider there are two binomial expression.
(a + b) and (m + n)

Lets us multiply both the expression using FOIL method.
(a + b) (m + n)

Given below are 4 steps as per FOIL method.

(a) F = First
Multiply First terms of both the binomial

(b) O = Outer
Multiply the outer terms present at the end of the whole expression.

(c) I = Inner
Multiply the terms present in inner side of whole expression.

(d) L = Last
Multiply the last terms of both the binomials

This is a 4 step method to multiply two binomials.
The acronym FOIL is very helpful to remember the exact steps needed to multiply the expressions.

Let us see some examples for further understanding.

## FOIL Method – Solved Examples

(01) Multiply the below binomials.
\mathtt{( x\ +\ xy) \ \&\ \left( x^{3} +5\right)}

Solution
We have to multiply the given binomials.
The expression can be written as:

\mathtt{\Longrightarrow \ ( x\ +\ xy) \ \left( x^{3} +5\right)}

Let’s follow the four steps of FOIL method

(i) F = First
Multiply the first terms of both the binomial

(ii) O = Outer
Multiply the outer term of combined expression

(iii) I = Inner
Multiply the inner terms of combined expression.

(iv) L = Last
Multiply last terms of both the binomials.

Hence, \mathtt{\ x^{4} +\ 5x\ +x^{4} y+5xy} is the solution.

Example 02
Multiply the below binomials.
\mathtt{\ ( 3x-10) \ \&\ ( x+7)}

Solution
The binomials can be written as:
\mathtt{\Longrightarrow \ ( 3x-10) \ ( x+7)}\

Now do the four steps of FOIL method.

(i) F = First
Multiply first terms of both binomials.

(ii) O = Outer
Multiply outer terms of the complete expression.

(iii) I = Inner
Multiply inner terms of the whole expression

(iv) L = Last
Multiply the last terms of both binomial

Hence, on multiplication we get;
\mathtt{\Longrightarrow \ 3x^{2} +\ 21x\ -10x-70}

The expression can be further simplified by subtracting the like terms.

\mathtt{\Longrightarrow \ 3x^{2} +\ 11x-70}

So, the above expression is the solution.

Example 03
Multiply the given binomials;
\mathtt{-5x+y) \ \&\ \left( x^{2} -11\right)}

Solution
The above expression can be written as;
\mathtt{\Longrightarrow \ ( -5x+y) \ \left( x^{2} -11\right)}

Now applying the four step FOIL method to multiply the binomials.

(i) F = First
Multiply the first term of both the binomial

(ii) O = Outer
Multiply the outer terms of both the binomial

(iii) I = Inner
Multiply inner terms of complete expressions

(iv) L = Last
Multiply last terms of both binomial

Hence, \mathtt{5x^{3} +\ 55x+x^{2} y\ -11y} is the solution.

Example 04
Multiply the binomials.
\mathtt{( -x\ -\ 3) \ ( -x\ +\ 2)}

Solution
The binomial can be written as:
\mathtt{( -x\ -\ 3) \ ( -x\ +\ 2)}

Now applying the four step FOIL method.

(i) F = First
Multiply the first term of both binomial

(ii) O = Outer
Multiply the outer parts of whole expression.

(iii) I = Inner
Multiply the inner terms of whole expression

(iv) L = Last
Multiply the last term of both binomial

Hence, on multiplication we get;
\mathtt{\Longrightarrow \ x^{2} -2x+3x-6}

Adding the similar terms, we get;

\mathtt{\Longrightarrow \ x^{2} +x-6}

Example 05
Multiply the below given polynomials
\mathtt{( -15+\ x) \ \&\ \left( x^{3} \ -\ 2\right)}

Solution
The expression can be written as;
\mathtt{( -15+\ x) \ \left( x^{3} \ -\ 2\right)}

Applying the FOIL method of binomial multiplication.

(i) F = First
Multiply the first term of both binomials

(ii) O = Outer
Multiply the outer terms of combined expression

(iii) I = Inner
Multiply the inner terms of combined expression

(iv) L = Last
Multiply the last terms of both binomials

Hence, \mathtt{-15x^{3} +30\ +\ x^{4} -2x} is the solution.

Example 06
Multiply the binomials using FOIL method
\mathtt{\left( x^{3} y+\ xy^{2}\right) \&\ ( xy+\ 3y)}

Solution
The expression can be written as:
\mathtt{\left( x^{3} y+\ xy^{2}\right)( xy+\ 3y)}

Multiply the binomials using 4 step FOIL method.

(i) F = First
Multiply the first term of binomials

(ii) O = Outer
Multiply the outer term of whole expression.

(iii) I = Inner
Multiply the inner terms of both expression

(iv) L = Last
Multiply the last term of both binomial

Hence, the above expression is the final solution.

## Frequently asked Question – FOIL method

(01) What is the full form of FOIL method?

(02) Can we use Foil method to multiply binomial with trinomial?

NO!!
FOIL method is only used to multiply two binomials.

(03) How is binomial different to Trinomial?

Binomial is an algebraic expression containing two entity.

Examples of binomial
\mathtt{\Longrightarrow \ x+\ y}\\\ \\ \mathtt{\Longrightarrow \ 2x^{2} +3}

While trinomial is an algebraic expression with three entity.

Examples of Trinomial
\mathtt{\Longrightarrow \ x+3\ +2y}\\\ \\ \mathtt{\Longrightarrow \ 6x+9xy+4}