2.5 The Distributive Property

Introduction

Suppose a store is offering \(10\%\) off every item in stock. This discount could be applied two different ways: either by reducing the price of each item individually or by calculating the total price of everything and then applying the discount.

As you probably know from experience, both methods give the same final result. What you might not know, however, is why the results are the same. That's what we'll explore in this section.

The Distributive Property

It's important to point out that the example in the introduction is an example of a special property that doesn't hold true for all mathematical expressions. In most cases, we can't simply rearrange the order of addition and multiplication without changing the outcome.

However, as the example illustrates, in the special case where the same number is multiplying several different values, there are two equivalent ways to evaluate the expression. This is relationship called the distributive property.

Many people find it helpful to draw arrows to indicate the multiplications that flow from applying the distributive property.

The distributive property also holds for subtraction: \(a(b-c)=a \cdot b - a \cdot c\) and can be extended to have multiple values inside the parenthesis including variables. It's also possible to distribute not just single numbers but entire expressions! We'll walk through examples of all of those below.

Distributing Numbers

Consider the expression \(3(4-9)\). Let's evaluate it the standard way and also using the distributive property, just to show the answers are the same.

Option 1. Use the order of operations:

\begin{align} 3(4-9) &= 3(-5) \\ &= -15 \end{align}

Option 2. Use the distributive property:

\begin{align} 3(4-9) &= 3(4)-3(9) \\ &= 12-27 \\ &= -15 \end{align}

When distributing an integer, be extra careful with any negative signs.

\begin{align} -2(-3+5-11) &= -2(-3)-2(5)-2(-11)\\ &= 6 - 10 + 22\\ &= 18 \end{align}

While using distribution might not offer a clear advantage over the standard order of operations when dealing with only numbers, its power becomes evident when expressions within parentheses involve variables. Consider the example:

\begin{align} -2.5(3x-4) &= -2.5(3x)-(-2.5)(4) \\ &= -7.5x-(-10) \\ &= -7.5x+10 \end{align}

In this case, distributing allows us to simplify the expression by applying the multiplication to term individually.

Distributing Monomials

By combining distribution with the product rule of exponents we can distribute monomials. A monomial is a single term containing a coefficient and possibly one or more variables raised to non-negative integer powers. For instance, \(-2x\) and \(3x^2\) are monomials.

Multiplying monomials involves multiplying the number parts (the coefficients) and multiplying the variable parts (using the product rule). To multiply \(-2x\) and \(3x^2\) we would multiply \(-2 \cdot 3=-6\) and multiply \(x \cdot x^2 = x^3\), yielding the answer \(-6x^3\).

When distributing monomials, we follow the same principle as before: each term inside the parentheses is multiplied by the monomial outside. Here's an example.

\begin{align} 3x(-2x+5) &= 3x(-2x)+3x(5) && \small \color{#5fa2ce} {\text{Distribute }3x} \\ &= -6x^2+15x && \small \color{#5fa2ce} {\text{Multiply the monomials}} \end{align}

Since distribution holds no mater how many terms are in the parenthesis, in this next example we'll distribute a monomial over an expression with three terms.

\begin{align} -4x^2(5x^5-7x+1) &= -4x^2(5x^5)-4x^2(-7x)-4x^2(1) && \small \color{#5fa2ce} {\text{Distribute }-4x^2} \\ &= -20x^7 + 28x^3 -4x^2 && \small \color{#5fa2ce} {\text{Multiply the monomials}} \end{align}

In these examples we've been distributing a monomial to a combination of numbers and variables within parentheses. But what if the monomial was replaced with a larger expression? Could we use the same approach? Let's find out.

Distributing Binomials

A binomial is an expression containing two monomial terms. For instance, \(3x-4\) and \(x+6\) are both binomials.

If we need to find the product \((3x-4)(x+6)\) we can apply the same process we used with monomials, but this time each term in the first binomial gets distributed to the entire second binomial, essentially making this a double distribution. The process will look like this:

Let's walk through those steps and see what we end up with.

\begin{align} (3x-4)(x+6) &= 3x(x+6) -4(x+6) && \small \color{#5fa2ce} {\text{Distribute each term to the second binomial}} \\ &= 3x^2+18x-4x-24 && \small \color{#5fa2ce} {\text{Distribute }3x \text{ and }-4} \\ &= 3x^2+14x-24 && \small \color{#5fa2ce} {\text{Combine like terms}} \\ \end{align}

To shorten the process, it can be helpful to draw arrows indicating all of the multiplications that need to be done. Here's the general template:

You might recognize this as the so-called "FOIL method". FOIL is a popular acronym some people find useful for remembering the steps for multiplying two binomials. It is called FOIL because we multiply the First terms, the Outer terms, the Inner terms, and then the Last terms of each binomial.

Since the FOIL method comes from the distributive property, if you are comfortable with distribution then it is not necessary to memorize what each FOIL letter stands for, but please use it if you find it helpful.

In our final example, we will multiply a binomial \(x+2\) by a trinomial \(3x^2-7x+6\). There is no FOIL acronym for this situation, but drawing distribution arrows can certainly help you keep track of all the multiplications regardless of the size of the two expressions.

With that as our guide, let's finish this example.