The distributive property can be used to simplify algebraic expressions. With the distributive property, we can distribute the term that is being multiplied by a parenthesis. In this article, we will look at a summary of the distributive property of algebraic expressions.
In addition, we will look at several examples with answers to fully master this topic. We will also see interactive problems to solve.
Summary of the distributive property
In algebra, we use the distributive property to remove parentheses when we simplify expressions. The distributive property tells us that we can distribute a term that is multiplying to a parenthesis that contains several terms:
For example, if we want to simplify the expression $latex 5(x+2)$, the order of operations tells us that we must solve the operations inside the parentheses first. However, we cannot add x and 2 since they are not like terms. So, we use the distributive property:
$latex 5(x+2)$
$latex =5(x)+5(2)$
$latex =5x+10$
It is also important to remember that when we simplify algebraic expressions, we are combining like terms. Like terms are terms of an expression that have the same variable raised to the same power. For example, $latex 3x$ and $latex 2x$ are like terms, and $latex {{x}^2}$ and $latex 5{{x}^2}$ are like terms as well.
Distributive property – Examples with answers
Each of the following distributive property examples has its respective solution. It is recommended that you try to solve the exercises yourself before looking at the solution. The detailed solution can be used to master the distributive property completely.
EXAMPLE 1
Simplify the expression $latex 10(x+3)$.
Solution
We use the distributive property to distribute the 10 to the terms inside the parentheses:
$latex 10(x)+10(3)$
Now, we multiply this and simplify:
$latex 10x+30$
EXAMPLE 2
Simplify the expression $latex 4x(2x+4)$.
Solution
We use the distributive property to distribute the 4x:
$latex 4x(2x)+4x(4)$
Now, we multiply and simplify:
$latex 8{{x}^2}+16x$
EXAMPLE 3
Simplify the expression $latex \frac{3}{2}(x+6)$.
Solution
We start by distributing the fraction $latex \frac{3}{2}$ to the terms inside the parentheses:
$latex \frac{3}{2}(x)+\frac{3}{2}(6)$
Now, we multiply and simplify the expression:
$latex \frac{3}{2}x+9$
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EXAMPLE 4
Simplify the expression $latex 5x(5x-6y)$.
Solution
Here we have two variables, x, and y. However, the procedure is the same, we just have to distribute the 5x to the terms inside the parentheses:
$latex 5x(5x)+5x(-6y)$
Now, we multiply and simplify:
$latex 25{{x}^2}-30xy$
EXAMPLE 5
Simplify the expression $latex -5y(3x-3y)$.
Solution
We distribute the -5y to the terms inside the parentheses without forgetting the sign change produced by the minus sign:
$latex -5y(3x)-5y(-3y)$
Now, we just have to multiply and simplify:
$latex -15xy+15{{y}^2}$
EXAMPLE 6
Simplify the expression $latex 4{{x}^2}(3x+4y+5)$.
Solution
Here, we expand the $latex 4{{x}^2}$ to the three inner terms:
$latex 4{{x}^2}(3x)+4{{x}^2}(4y)+4{{x}^2}(5)$
Now, we multiply and simplify:
$latex 12 {{x}^3}+16{{x}^2}y+20{{x}^2}$
EXAMPLE 7
Simplify the expression $latex 2x(5{{x}^3}+3{{x}^2}+5x)$.
Solution
We distribute the 2 x to the three interior terms:
$latex 2x(5{{x}^3})+2x(3{{x}^2})+2x(5x)$
We multiply and simplify:
$latex 10{{x}^4}+6{{x}^3}+10{{x}^2}$
Distributive property – Practice problems
The following distributive property problems can be used to test your knowledge on this topic. Choose an answer and click “Check” to find the correct solution. You can look at the examples with answers above if you have problems with this topic.
See also
Interested in learning more about algebraic expressions? Take a look at these pages:
