Explainer: Multiplying Binomials

In this explainer, we will learn how to multiply binomials with integer and fractional coefficients.

First, we will start by understanding binomials.

Definition

A binomial is an expression that contains the sum or difference of two terms.

If you are confident with the definition of a polynomial, you can define a binomial as a polynomial that contains two terms.

Here are some examples of binomials:

  • 3π‘₯+5π‘₯
  • π‘₯+𝑦
  • 7π‘₯βˆ’5𝑦
  • π‘₯βˆ’7

When we talk about multiplying binomials, we are talking about finding the product of two, three, or more binomials that could be the same or different. There are a number of strategies that you can use to complete this process and we will look at a few of these here.

Before we look at multiplying two binomials, let us first look at multiplying a monomial (a single term expression) and a binomial. For example, we may be asked to expand 6π‘₯(5π‘₯π‘¦βˆ’3). This means 6π‘₯Γ—5π‘₯𝑦+6π‘₯Γ—(βˆ’3). If we simplify each of these multiplications, we get 30π‘₯π‘¦βˆ’18π‘₯,which is the result of our expansion.

Now, let us look at multiplying two binomials. This is a very similar process to multiplying a binomial by a monomial. Consider expanding (π‘₯+3)(π‘₯+2). This can be rewritten as π‘₯(π‘₯+2)+3(π‘₯+2).

If we expand each of the brackets separately, we get π‘₯+2π‘₯+3π‘₯+6, which simplifies to π‘₯+5π‘₯+6.

Let us look at an example.

Example 1: Multiplying Two Binomials

Expand and simplify (βˆ’2π‘₯βˆ’4𝑦)(2π‘₯βˆ’π‘¦).

Answer

First, let us rewrite the expression βˆ’2π‘₯(2π‘₯βˆ’π‘¦)βˆ’4𝑦(2π‘₯βˆ’π‘¦).

If we then expand each of the brackets, being very careful when multiplying negative numbers, we get βˆ’4π‘₯+2π‘₯π‘¦βˆ’8π‘₯𝑦+4𝑦.

Pay particular attention to the β€œ+2π‘₯𝑦” and the β€œ+4π‘¦οŠ¨β€ as the most common errors are to write these as β€œβˆ’2π‘₯𝑦” and β€œβˆ’4π‘¦οŠ¨β€ respectively. We then need to simplify the resulting expression by collecting like terms: βˆ’4π‘₯βˆ’6π‘₯𝑦+4𝑦.

Let us now look at an example where you multiply a binomial by itself.

Example 2: Multiply a Binomial by Itself

Simplify (3π‘₯+4).

Answer

First, recall that (3π‘₯+4) is the same as (3π‘₯+4)(3π‘₯+4). We can then rewrite this expression as 3π‘₯(3π‘₯+4)+4(3π‘₯+4). If we expand each bracket, we get 9π‘₯+12π‘₯+12π‘₯+16, which we can then simplify by collecting like terms: 9π‘₯+24π‘₯+16.

In addition to the methods used in these first examples, there are a number of other methods that can be used to multiply binomials, which we will demonstrate in another couple of examples. The first of these is the area model, which is also referred to as the grid method.

Example 3: Multiplying Binomials Using the Area Model (or Grid Method)

Expand and simplify ο€Ό2π‘Ž3βˆ’7π‘οˆο€Ό5π‘Ž3βˆ’8π‘οˆ.

Answer

We can calculate this using the area model. We can rewrite the expansion in a way that represents the calculation of a rectangle’s area as follows:

We can then work out the value or β€œarea” of the subrectangles starting with the top left:

Then, we work out the value of the top right subrectangle:

The bottom left subrectangle is next:

Finally, we find the value of the bottom right one:

In order to find the final expansion, we sum together all of the values, collecting any like terms. Here, our like terms are βˆ’35π‘Žπ‘3 and βˆ’16π‘Žπ‘3. So, our final answer is 10π‘Ž9βˆ’17π‘Žπ‘+56𝑏.

An alternative method for multiplying is to use arrows to simplify your working. This is sometimes referred to as the β€œlobster claw method”. We will demonstrate this now.

Example 4: Multiplying Binomials Using Arrows (or the Lobster Claw Method)

Expand (βˆ’π‘₯+2𝑦).

Answer

First, remember that (βˆ’π‘₯+2𝑦)=(βˆ’π‘₯+2𝑦)(βˆ’π‘₯+2𝑦). We can then draw arrows to remind us that we first need to multiply βˆ’π‘₯ by (βˆ’π‘₯+2𝑦) and then we need to add the multiple of 2𝑦 and (βˆ’π‘₯+2𝑦) as follows:

Dealing with each arrow separately, we have: (βˆ’π‘₯)Γ—(βˆ’π‘₯)=+π‘₯,(βˆ’π‘₯)Γ—(2𝑦)=βˆ’2π‘₯𝑦,(2𝑦)Γ—(βˆ’π‘₯)=βˆ’2π‘₯𝑦,(2𝑦)Γ—(2𝑦)=4𝑦.

Simplifying by collecting like terms, we get π‘₯βˆ’4π‘₯𝑦+4𝑦.

It is a good idea to choose the method with which you are most comfortable and use that. One final example that we will look at is what to do if you are multiplying three binomials. The general method is to start by multiplying the first two and then multiply the result of this by the third. Let us look at this now.

Example 5: Multiplying Three Binomials

Expand and simplify (4π‘₯+𝑦).

Answer

Remember that this means (4π‘₯+𝑦)(4π‘₯+𝑦)(4π‘₯+𝑦). Our first step is to multiply out two of the brackets first; in other words, expand (4π‘₯+𝑦)(4π‘₯+𝑦): (4π‘₯+𝑦)(4π‘₯+𝑦)=16π‘₯+4π‘₯𝑦+4π‘₯𝑦+𝑦.

This simplifies to 16π‘₯+8π‘₯𝑦+𝑦.

We then need to multiply this by another (4π‘₯+𝑦) (you may want to draw arrows to help avoid missing out any terms): (4π‘₯+𝑦)ο€Ή16π‘₯+8π‘₯𝑦+𝑦=64π‘₯+32π‘₯𝑦+4π‘₯𝑦+16π‘₯𝑦+8π‘₯𝑦+𝑦.

Simplifying, we get 64π‘₯+48π‘₯𝑦+12π‘₯𝑦+𝑦.

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