Algebraic Expressions

Introduction

Algebraic expressions are made up of terms. A term is a constant or the product of a constant and one or more variables. Some examples of terms are $7,y,5x^{2},9a,$ and $13xy$. The constant that multiplies the variable(s) in a term is called the coefficient. We can think of the coefficient as the number in front of the variable. For example, the coefficient of the term 3x is 3. When we write x, the coefficient is 1, since x = 1(x). To evaluate an algebraic expression means to find the value of the expression when the variable is replaced by a given number. To evaluate an expression, we substitute the given number for the variable in the expression, and then simplify the expression using the order of operations.

Some terms share common traits. Terms are called like terms if they have the same variables and exponents. All constant terms are also like terms. Consider the terms $5x,7,n^{2},4,3x,9n^{2},-2xy$, and $8xy$. From this list, 7 and 4 are like terms, 5x and 3x are like terms, $-2xy$ and $8xy$ are like terms, and $n^{2}$ and $9n^{2}$ are like terms.

We can simplify an expression by combining the like terms. To do this, we add the coefficients and keep the same variable. For example, say we want to simplify the expression $x^{2}+4x^{2}-3x+6+x$. The coefficients of the like terms $x^{2}$ and $4x^{2}$ are 1 and 4, and add up to 5. The coefficients of the like terms -3x and x are 1 and -3, and add to -2. 6 is not like to any of the other terms, so the constant term remains as 6. So, we can combine like terms and get that $x^{2}+4x^{2}-3x+6+x=(x^{2}+4x^{2})+(-3x+x)+6=5x^{2}-2x+6$ by adding up the coefficients of like terms.

Expressions and Terms

All algebraic equations can be considered as a series of expressions. On each side of the equals '=' sign there will be an expression. For example:

$5x+2x=7$

contains the expressions $(5x+2x)$ and $(7)$

Within each expression, there will be a number of terms. A term is any variable, number, or algebraic letter. In the equation above we can say that there are three terms. These three terms are:

$5x$ $2x$ and $7$

Now what is key to all equations is simplifying what they say, that is reducing the length of the expression by collecting like terms together. We have seen that the equation above technically has three terms. Let us break down a term further.

The term $5x$ has a coefficient of $5$ and a variable of $x$

When you simplify an expression, you simply collect the like terms together. In other words, you collect the terms with the same variables together.

So taking our very first equation: $5x+2x=7$ we can see that the terms $5x$ and $+2x$ contain the same variable $x$. They can therefore be added together to obtain $7x$

So, From above we can see that like terms can only be collected together in order to simplify an expression and hence an equation. Terms which do not contain identical variables cannot be simplified. The exaples below show collecting like and unlike terms together.

Example 1

Simplify the expression $5x^{2}+2x+3x+5+7$

The Table to the right shows the number of terms with each variable.

Variable	Number of Terms
$x^{2}$	1
$x$	2
No Variable	2

Reading this table we can see that:

• There is only 1 term in $x^{2}$ ($5x^{2})$) and so it cannot be simplified.
• There are two terms in $x$ ($+2x$ and $+3x$) so these can be added.
• There are two terms with no variable $+5$ and $+7$ and these can be added.

Therefore, after simplifying the expression we finish with $5x^{2}+5x+12$

Example 2

Simplify the expression $3x^{2}+2x+2y-4y-3x+x^{2}-xy$

Using the rules stated above the expression can be simplified to:

$4x^{2}-x-2y-xy$

Single Variables

In example 2 above we ended up with the term $-x$. It is important to note that when there is just a variable, with no apparent coefficient, the coefficient is in fact 1.

• $-x$ is the same as $-1x$
• $xy$ is the same as $1xy$

Resources

• Evaluate, Simplify, and Translate Expressions, OpenStax
• Evaluate and Simplify Algebraic Expressions, Lumen Learning
• Evaluating Algebraic Expressions, Palm Beach State College