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# Multiplying Binomial Expressions

by Ron Kurtus (revised 18 January 2022)

A * binomial expression* is an algebraic expression consisting of two terms or monomials separated by a plus (

**+**) or minus (

**−**) sign. Examples of binomials include:

**ax + b**,

**x**, and

^{2}− y^{2}**2x + 3y**.

In Algebra, you are often required to multiply expressions together. The best way to multiply two binomial expressions is to use what is called the **FOIL** method. In this method, you multiply the **F**irst, **O**utside, **I**nside, and **L**ast terms and then add them together.

You should follow some good practices, such as putting terms in the right order, before you start. Also, there are a few shortcuts that are good to remember.

Questions you may have include:

- What is the FOIL method?
- What are some good practices?
- What are some special situations?

This lesson will answer those questions.

## FOIL method

Since binomials are simple and you are smart, a **FOIL** method is usually used to multiply two binomials.

**FOIL** stands for: "multiply the **First** terms, multiply the **Outside** terms, multiply the **Inside** terms, and multiply the **Last** terms." Then you add the results together in the proper order.

For example, to multiply **(ax + b)(cx + d)**, you follow the procedure:

1. Multiply the two

Firstterms together(ax + b)(cx + d):(ax)(cx)=acx^{2}2. Multiply the two

Outsideterms(ax + b)(cx + d):(ax)(d) = adx3. Multiply the two

Insideterms(ax + b)(cx + d):(b)(cx) = bcx4. Multiply the two

Lastterms(ax + b)(cx + d):(b)(d) = bd5. Add the results to get:

acx^{2}+ adx + bcx + bdThis can also be written as:

acx^{2}+ (ad + bc)x + bd

Typically, you can do these operations in your head, writing down the results in their order.

## Good practices

There are different situations you can study.

### Arrange terms

It is a good practice to arrange the terms in a uniform order. For example, supposed you want to multiply **(3y + x)(2x − 5y)**. Although you will get the same answer using that order, it is best and easier to rearrange the terms as **(x + 3y)(2x − 5y)**.

Then multiply **(x + 3y)(2x − 5y)**:

1. First terms:

(x)(2x)=2x^{2}2.Outside terms:

(x)(−5y) = −5xy3. Inside terms:

(3y)(2x) = 6xy4. Last terms:

(3y)(−5y)=−15y^{2}5. Add together:

2x^{2 }−5xy + 6xy −15y^{2}6. Combine like terms to get the final result:

2x^{2}+ xy − 15y^{2}

Likewise, you should rearrange** (xa − 7)(5x + 2)** to be **(ax − 7)(5x + 2)** before multiplying.

### Simplify expressions

You also want to simplify expressions, if possible. For example, the expression **2x + 3 − 8** can be simplified into a binomial expression: **2x −5**.

## Special situations

There are special situations to be aware of.

### When binomials not similar

When the binomials are not similar, it can get tricky. For example, multiply **(x ^{2} − y)(x − 2y)**:

1. First:

(x=^{2})(x)x^{3}2.Outside:

(x^{2})(−2y) = −2x^{2}y3. Inside:

(−y)(x) = −xy(note that we changed the order ofxandy)4. Last:

(−y)(−2y)=2y^{2}5. Add together:

x^{3 }− 2x^{2}y− xy +2y^{2}

This result has four terms instead of the usual three terms.

### Squaring an expression

If you are going to multiply an expression by itself, such as **(ax + b) ^{2} = (ax + b)(ax + b)**, it is easy to get the result without the

**FOIL**method.

(ax + b)(ax + b) = a^{2}x^{2}+ 2abx + b^{2}

Also

(ax − b)(ax − b) = a^{2}x^{2}− 2abx + b^{2}

It is good to remember this shortcut.

### Special form

When you multiply binomials in the form of **(a + b)(a −b)**, the result is **a ^{2} − b^{2}**.

For example, multiply **(2x + 3y)(2x − 3y)**

1. First:

(2x)(2x)=4x^{2}2. Outside:

(2x)(−3y) = −6xy

3. Inside:

(3y)(2x) = 6xy4. Last:

(3y)(−3y)=−9y^{2}5. Add together:

4xfor the final result:^{2 }−6xy + 6xy −9y^{2}

4xor^{2}− 9y^{2}2^{2}x^{2}− 3^{2}y^{2}

Remember this shortcut, because it will come up time and time again in Algebra.

Another example is simply: **(x − 2)(x + 2) = x ^{2} − 4**

## Exercises

Try the following exercises:

1. **(5x − 7)(x + 2)**

2. **(x ^{2} + 3)(x^{2} + 3)**

3. **(x − y)(3x − 2y)**

4. **(4 − y)(y + 4)**

5. **(a + b)(c + d)**

### Answers

1. **5x ^{2} + 3x − 14**

2. **x ^{4} + 6x^{2} + 9**

3. **3x ^{2} − 5xy + 2y^{2}**

4. **16 − y ^{2}**

^{}

5. **ac + ad + bc + bd**

## Summary

You are often required to multiply expressions together. The best way to multiply two binomial expressions is to use what is called the **FOIL** method.

There are some good practices to follow, such as putting terms in the right order.

Some shortcuts to remember are:

(a + b)(a + b) = a^{2}+ 2ab + b^{2}

(a − b)(a − b) = a^{2}− 2ab + b^{2}

(a + b)(a − b) = a^{2}− b^{2}

Use clever tools to make work easier

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## Multiplying Binomial Expressions