Simplifying Algebraic Expressions
Learning Outcomes
- Identify the variables and constants in a term
- Identify the coefficient of a variable term
- Identify and combine like terms in an expression
Identify Terms, Coefficients, and Like Terms
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 [latex]7,y,5{x}^{2},9a,\text{and }13xy[/latex]. 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. The coefficient of the term [latex]3x[/latex] is [latex]3[/latex]. When we write [latex]x[/latex], the coefficient is [latex]1[/latex], since [latex]x=1\cdot x[/latex]. The table below gives the coefficients for each of the terms in the left column.Term | Coefficient |
---|---|
[latex]7[/latex] | [latex]7[/latex] |
[latex]9a[/latex] | [latex]9[/latex] |
[latex]y[/latex] | [latex]1[/latex] |
[latex]5{x}^{2}[/latex] | [latex]5[/latex] |
Expression | Terms |
---|---|
[latex]7[/latex] | [latex]7[/latex] |
[latex]y[/latex] | [latex]y[/latex] |
[latex]x+7[/latex] | [latex]x,7[/latex] |
[latex]2x+7y+4[/latex] | [latex]2x,7y,4[/latex] |
[latex]3{x}^{2}+4{x}^{2}+5y+3[/latex] | [latex]3{x}^{2},4{x}^{2},5y,3[/latex] |
example
Identify each term in the expression [latex]9b+15{x}^{2}+a+6[/latex]. Then identify the coefficient of each term. Solution: The expression has four terms. They are [latex]9b,15{x}^{2},a[/latex], and [latex]6[/latex].- The coefficient of [latex]9b[/latex] is [latex]9[/latex].
- The coefficient of [latex]15{x}^{2}[/latex] is [latex]15[/latex].
- Remember that if no number is written before a variable, the coefficient is [latex]1[/latex]. So the coefficient of [latex]a[/latex] is [latex]1[/latex].
- The coefficient of a constant is the constant, so the coefficient of [latex]6[/latex] is [latex]6[/latex].
try it
[ohm_question]144899[/ohm_question]- The terms [latex]7[/latex] and [latex]4[/latex] are both constant terms.
- The terms [latex]5x[/latex] and [latex]3x[/latex] are both terms with [latex]x[/latex].
- The terms [latex]{n}^{2}[/latex] and [latex]9{n}^{2}[/latex] both have [latex]{n}^{2}[/latex].
- [latex]7[/latex] and [latex]4[/latex] are like terms.
- [latex]5x[/latex] and [latex]3x[/latex] are like terms.
- [latex]{n}^{2}[/latex] and [latex]9{n}^{2}[/latex] are like terms.
Like Terms
Terms that are either constants or have the same variables with the same exponents are like terms.example
Identify the like terms:- [latex]{y}^{3},7{x}^{2},14,23,4{y}^{3},9x,5{x}^{2}[/latex]
- [latex]4{x}^{2}+2x+5{x}^{2}+6x+40x+8xy[/latex]
Answer: Solution: 1. [latex]{y}^{3},7{x}^{2},14,23,4{y}^{3},9x,5{x}^{2}[/latex] Look at the variables and exponents. The expression contains [latex]{y}^{3},{x}^{2},x[/latex], and constants. The terms [latex]{y}^{3}[/latex] and [latex]4{y}^{3}[/latex] are like terms because they both have [latex]{y}^{3}[/latex]. The terms [latex]7{x}^{2}[/latex] and [latex]5{x}^{2}[/latex] are like terms because they both have [latex]{x}^{2}[/latex]. The terms [latex]14[/latex] and [latex]23[/latex] are like terms because they are both constants. The term [latex]9x[/latex] does not have any like terms in this list since no other terms have the variable [latex]x[/latex] raised to the power of [latex]1[/latex]. 2. [latex]4{x}^{2}+2x+5{x}^{2}+6x+40x+8xy[/latex] Look at the variables and exponents. The expression contains the terms [latex]4{x}^{2},2x,5{x}^{2},6x,40x,\text{and}8xy[/latex] The terms [latex]4{x}^{2}[/latex] and [latex]5{x}^{2}[/latex] are like terms because they both have [latex]{x}^{2}[/latex]. The terms [latex]2x,6x,\text{and}40x[/latex] are like terms because they all have [latex]x[/latex]. The term [latex]8xy[/latex] has no like terms in the given expression because no other terms contain the two variables [latex]xy[/latex].
try it
[ohm_question]146540[/ohm_question]Simplify Expressions by Combining Like Terms
We can simplify an expression by combining the like terms. What do you think [latex]3x+6x[/latex] would simplify to? If you thought [latex]9x[/latex], you would be right! We can see why this works by writing both terms as addition problems. Add the coefficients and keep the same variable. It doesn’t matter what [latex]x[/latex] is. If you have [latex]3[/latex] of something and add [latex]6[/latex] more of the same thing, the result is [latex]9[/latex] of them. For example, [latex]3[/latex] oranges plus [latex]6[/latex] oranges is [latex]9[/latex] oranges. We will discuss the mathematical properties behind this later. The expression [latex]3x+6x[/latex] has only two terms. When an expression contains more terms, it may be helpful to rearrange the terms so that like terms are together. The Commutative Property of Addition says that we can change the order of addends without changing the sum. So we could rearrange the following expression before combining like terms. Now it is easier to see the like terms to be combined.Combine like terms
- Identify like terms.
- Rearrange the expression so like terms are together.
- Add the coefficients of the like terms.
example
Simplify the expression: [latex]3x+7+4x+5[/latex].Answer: Solution:
[latex]3x+7+4x+5[/latex] | |
Identify the like terms. | [latex]\color{red}{3x}+\color{blue}{7}+\color{red}{4x}+\color{blue}{5}[/latex] |
Rearrange the expression, so the like terms are together. | [latex]\color{red}{3x}+\color{red}{4x}+\color{blue}{7}+\color{blue}{5}[/latex] |
Add the coefficients of the like terms. | |
The original expression is simplified to... | [latex]7x+12[/latex] |
try it
[ohm_question]144900[/ohm_question]example
Simplify the expression: [latex]8x+7{x}^{2}-{x}^{2}-+4x[/latex].Answer: Solution:
[latex]8x+7{x}^{2}-{x}^{2}-+4x[/latex] | |
Identify the like terms. | |
Rearrange the expression so like terms are together. | |
Add the coefficients of the like terms. |
try it
[ohm_question]144905[/ohm_question]Licenses & Attributions
CC licensed content, Original
- Simplify Expressions by Combining Like Terms (No Negatives). Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. License: CC BY: Attribution.
CC licensed content, Shared previously
- Ex 1: Combining Like Terms. Authored by: James Sousa (Mathispower4u.com). License: CC BY: Attribution.
- Ex 2: Combining Like Terms. Authored by: James Sousa (Mathispower4u.com). License: CC BY: Attribution.
- Question ID: 144899, 144900, 144905,146540. Authored by: Alyson Day. License: CC BY: Attribution. License terms: IMathAS Community License CC-BY + GPL.
CC licensed content, Specific attribution
- Prealgebra. Provided by: OpenStax License: CC BY: Attribution. License terms: Download for free at http://cnx.org/contents/[email protected].