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Simplifying Using the Distributive Property Lesson

The Distributive Property is an algebra property which is used to multiply a single term and two or more terms inside a set of parentheses. Take a look at the problem below.

2(3 + 6)

Because the binomial "3 + 6" is in a set of parentheses, when following the Order of Operations, you must first find the answer of 3 + 6, then multiply it by 2. This gives an answer of 18.

2(3 + 6)
2(9)
18

! Incorrect Method !

It would be incorrect to remove the parentheses and multiply 2 and 3 then add 6, as this would give an incorrect answer of 12.

2(3 + 6)
2 * 3 + 6
6 + 6
12

Examine the expression below.

6(2+4x)

The two terms inside the parentheses cannot be added because they are not like terms. Therefore, 2 + 4x, the expression inside the parentheses, cannot be simplified any further. To simplify this multiplication, another method will be needed. This is where the Distributive Property comes in.

Distributing a Number

We continue with previous example.

6(2 + 4x)

The Distributive Property tells us that we can remove the parentheses if the term that the polynomial is being multiplied by is distributed to, or multiplied with each term inside the parentheses.

This definition is tough to understand without a good example, so observe the example below carefully.

6(2 + 4x)

now by applying the Distributive Propery

6 * 2 + 6 * 4x

The parentheses are removed and each term from inside is multiplied by the six.

Now we can simplify the multiplication of the individual terms:

12 + 24x

Distributing a Negative Sign

The next problem does not have a number outside the parentheses, only a negative sign.

-(3 + x2)

There are two easy ways to simplify this problem. The first and simplest way is to change each positive or negative sign of the terms that were inside the parentheses. Negative or minus signs become positive or plus signs. Similarly, positive or plus signs become negative or minus signs. Recall that in the case of 3, no positive or negative sign is shown, so a positive sign is assumed.

-3 - x2

We will now work through this problem again, but using a different method.

-(3 + x2)

Recall that any term that does not have a coefficient has an implied coefficient of 1. Because of the negative sign on the parentheses, we instead assume a coefficient of negative one. Thus, we can rewrite the problem as

-1(3 + x2)

Now the -1 can be distributed to each term inside the parentheses as in the first example in this lesson.

-1 * 3 + -1 * x2
-3 + -x2

Distributing Variables

A variable can be distributed into a set of parentheses just as we distributed a negative sign or a number. Consider the following example.

x(y + 1)

We can now apply the distributive property to the expression by multiplying each term inside the parentheses by x.

x * y + x * 1

Now simplifying the multiplication, we get a final answer of

xy + x

The same is true when a problem consists of a number, variables, and parentheses:

4x(x2 + 9)

Again, multiply each term inside the parentheses by the multiplier outside the parentheses.

4x * x2 + 4x * 9

Then simplify

4x3 + 36x

Quiz on the Distributive Property

1. Solve the following expression

A.
B.
C.
D.
The correct answer here would be B.

The outside term distributes evenly into the parentheses i.e. it multiplies both terms

2. Evaluate the following without using a calculator

A.
B.
C.
D.
The correct answer here would be D.

The distributive property allows for these two numbers to be multiplied by breaking up the larger one into a sum of smaller ones and then applying the property as shown below

Therefore:

The above multiplications are relatively easier:

Distributive Property Resources

Practice Problems / Worksheet
Practice applying the Distributive Property with these expressions.

Expression Simplifying Calculator
This calculator will simplify expressions, applying the distributive property when necessary.

Next Lesson: FOIL Method
Using the FOIL Method to multiply two or more parenthesis. (Multiplying two Binomials, or two Polynomials)

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