# Basic Algebra/Factoring/Squares of Binomials

## Contents

## Vocabulary[edit]

__Binomial__ - An algebraic expression with exactly two terms.

__Square__ - Multiply a number by itself.

__FOIL__ - The product of two binomials is the sum of the products of the First terms, the Outer terms, the Inner terms, and the Last terms.

__Quantity__ -Total amount or number.

## Technical Equation[edit]

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

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

__NOTE__: These equations will work if you substitute the first term for **a** and the second term for **b**. I find these equations easier to understand after the concept is learned using the three steps described below.

## Lesson[edit]

Lesson 4 has shown you how to multiply binomials. In Lesson 5 we are going to learn how to square binomials. Squaring a binomial can be done using two different methods. The first method uses FOIL (refer to lesson 4). The second method is a shorter alternative to FOIL. The way we use the shortcut is to follow three simple steps.

__Step 1:__ Square the first term of the binomial.

__Step 2:__ Multiply the first term and last term of the binomial together and then double that quantity (in other words multiply by 2).

__Step 3:__ Square the last term of the binomial.

## Example Problems[edit]

**Here is an example to follow.**

Using the binomial (x+6) we will square it creating the problem (x+6)^{2}

__Step 1__ Square the first term of the binomial.

(x)^{2}=x^{2}

__Step 2__ Multiply the first term and last term of the binomial together and then double that quantity (in other words multiply by 2).

[(x)*(6)]*2 = (6x)*(2) = 12x

__Step 3__ Square the last term of the binomial.

(6)^{2} = 36

Finally we put the three term we have acquired together and get the answer

(x+6)^{2} = x^{2} + 12x + 36

**Let’s try one more problem that may be a more difficult.**

Let’s square the binomial (x^{2}-4x) giving us (x^{2}-4x)^{2}

__Step 1__ Square the first term of the binomial.

(x^{2})^{2} = x^{4}

__Step 2__ Multiply the first term and last term of the binomial together and then double that quantity (in other words multiply by 2).

*Notice we keep the negative sign with the second term

[(x^{2})(-4x)]*2 = (-4x^{3})*(2) = -8x^{3}

__Step 3__ Square the last term of the binomial.

(-4x)^{2} = (-4)^{2}(x)^{2} = 16x^{2}

Our final answer will be the answers from the three steps combined

(x^{2}-4x)^{2} = x^{4} -8x^{3} + 16x^{2}

## Online Lesson[edit]

You can use this link to view this lesson illustrated by a teacher.

http://www.phschool.com/atschool/academy123/html/bbapplet_wl-problem-431067.html

(2x-6y)2 = 2x2 - 6xy + 6y2

## Practice Problems w/Answers[edit]

(x+4)^{2} = x^{2}+ 8x+ 16

(x+y)^{2}= x^{2}+ 2xy + y^{2}

(2x^{2} + 3y^{2})^{2} = 4x^{4} + 12x^{2}y^{2} + 9y^{4}

## Note to Instructor[edit]

Students may have trouble transitioning from the FOIL method to the shortcut. I found that saying the steps out loud every time will help students remember the steps. This process can cut down their problem solving time as they progress through their math courses.