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}