We can use the simplification of algebraic expressions to get simpler expressions to work with. You can simplify expressions by applying the distributive property of multiplication to remove grouping signs such as parentheses and combine like terms.
In this article, we will look at a summary of simplifying algebraic expressions. In addition, we will explore several simplification examples with answers to fully master this topic.
Summary of simplification of algebraic expressions
The simplification of algebraic expressions allows us to obtain simpler expressions that can be manipulated with ease. To simplify algebraic expressions, we can apply the distributive property to remove parentheses and other grouping signs, and we can combine like terms.
Distributive property
The distributive property tells us how to eliminate grouping signs by distributing the multiplication of a number to all the internal terms of the parentheses:
Combine like terms
Like terms are algebraic terms that have the same variables raised to the same power. For example, $latex 2{{x}^2}$ and $latex 3{{x}^2}$ are like terms since they have the same variable (x) raised to the same power (2). Similarly, the terms $latex 3x{{y}^3}$ and $latex 2x{{y}^3}$ are also like terms since they have the same variables raised to the same power.
Simplification of algebraic expressions – Examples with answers
The following examples have their respective solutions, so you can study them carefully and master the process of simplifying algebraic expressions.
EXAMPLE 1
Simplify the algebraic expression: $latex 2x+4+3x-5$.
Solution
We have the variable x and we have constant terms. Therefore, we combine with terms with variables and combine the constant terms:
$latex 2x+4+3x-5$
$latex =(2x+3x)+(4-5)$
$latex =5x-1$
EXAMPLE 2
Simplify the algebraic expression: $latex 3x+2(3x-2)+10$.
Solution
We have a parenthesis so we start by using the distributive property to distribute the 2 and remove the parentheses:
$latex 3x+2(3x-2)+10$
$latex =3x+6x-4+10$
Now we combine like terms. We combine the variables and the constant terms:
$latex =(3x+6x)+(-4+10)$
$latex =9x+6$
EXAMPLE 3
Simplify the algebraic expression: $latex 4x+2{{x}^2}+5-3x+4{{x}^2}+4$.
Solution
In this case, we have the variable x with a power of 1 and with a power of 2, so we combine like terms separately for the power of 1 and for the power of 2. We also combine the constant terms separately:
$latex 4x+2{{x}^2}+5-3x+4{{x}^2}+4$
$$=(2{{x}^2}+4{{x}^2})+(4x-3x)+(5+4)$$
$latex =6{{x}^2}+x+9$
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EXAMPLE 4
Simplify the algebraic expression: $latex 3x(2x+5)+10x-6+5{{x}^2}$.
Solution
We start by applying the distributive property to remove the parentheses:
$latex 3x(2x+5)+10x-6+5{{x}^2}$
$latex =6{{x}^2}+15x+10x-6+5{{x}^2}$
Now combine like terms. We combine terms with the variable x with different powers separately:
$latex =(6{{x}^2}+5{{x}^2})+(15x+10x)-6$
$latex =11{{x}^2}+25x-6$
EXAMPLE 5
Simplify the algebraic expression: $latex 4x(y+3x)-3xy+5{{x}^2}+10-6x$.
Solution
We start by removing the parentheses using the distributive property:
$latex 4x(y+3x)-3xy+5{{x}^2}+10-6x$
$$=4xy+12{{x}^2}-3xy+5{{x}^2}+10-6x$$
In this case, we have terms with both the x variable and the y variable. We have to combine terms that have the same variables raised to the same powers:
$$=(4xy-3xy)+(12{{x}^2}+5{{x}^2})+10-6x$$
$latex =xy+17{{x}^2}-6x+10$
EXAMPLE 6
Simplify the algebraic expression: $$5(2x^2y+4x)+3x-2x(4xy-5)+12$$
Solution
We start by removing both parentheses using the distributive property:
$$5(2x^2y+4x)+3x-2x(4xy-5)+12$$
$$=10{{x}^2}y+20x+3x-8{{x}^2}y-10x+12$$
We have to combine the terms that have the same variables raised to the same power:
$$=(10{{x}^2}y-8{{x}^2}y)+(20x+3x-10x)+12$$
$latex =2{{x}^2}y+13x+12$
EXAMPLE 7
Simplify the algebraic expression: $$-2x(3{{x}^2}+2x-1)+4x+2{{x}^2}+6-4{{x}^3}-10x+5$$
Solution
We start by removing the parentheses using the distributive property:
$latex -2x(3{{x}^2}+2x-1)+4x+2{{x}^2}$ $latex +6-4{{x}^3}-10x+5$
$latex =-6{{x}^3}-4{{x}^2}+2x+4x+2{{x}^2}$ $latex +6-4{{x}^3}-10x+5$
Here we have several terms with the variable x with different powers. We have to make sure to combine only the terms that have the same power:
$latex =(-6{{x}^3}-4{{x}^3})+(-4{{x}^2}+2{{x}^2})$ $latex +(2x+4x-10x)+(5+6)$
$latex =-10{{x}^3}-2{{x}^2}-4x+11$
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Simplifying algebraic expressions – Practice problems
In addition to the solved examples, the following problems help to master the simplification of algebraic expressions. If you have trouble solving these exercises, you can carefully study the solved examples above.
See also
Interested in learning more about algebraic expressions? Take a look at these pages:
