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Identifying Terms, Coefficients, and Like Terms

Identifying 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 \(7,y,5{x}^{2},9a,\text{and}\phantom{\rule{0.2em}{0ex}}13xy.\)

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. The coefficient of the term \(3x\) is \(3.\) When we write \(x,\) the coefficient is \(1,\) since \(x=1\cdot x.\) The table below gives the coefficients for each of the terms in the left column.

TermCoefficient
\(7\)\(7\)
\(9a\)\(9\)
\(y\)\(1\)
\(5{x}^{2}\)\(5\)

An algebraic expression may consist of one or more terms added or subtracted. In this tutorial, we will only work with terms that are added together. The table below gives some examples of algebraic expressions with various numbers of terms. Notice that we include the operation before a term with it.

ExpressionTerms
\(7\)\(7\)
\(y\)\(y\)
\(x+7\)\(x,7\)
\(2x+7y+4\)\(2x,7y,4\)
\(3{x}^{2}+4{x}^{2}+5y+3\)\(3{x}^{2},4{x}^{2},5y,3\)

Example

Identify each term in the expression \(9b+15{x}^{2}+a+6.\) Then identify the coefficient of each term.

Solution

The expression has four terms. They are \(9b,15{x}^{2},a,\) and \(6.\)

The coefficient of \(9b\) is \(9.\)

The coefficient of \(15{x}^{2}\) is \(15.\)

Remember that if no number is written before a variable, the coefficient is \(1.\) So the coefficient of \(a\) is \(1.\)

The coefficient of a constant is the constant, so the coefficient of \(6\) is \(6.\)

Some terms share common traits. Look at the following terms. Which ones seem to have traits in common?

\(5x,7,{n}^{2},4,3x,9{n}^{2}\)

Which of these terms are like terms?

  • The terms \(7\) and \(4\) are both constant terms.
  • The terms \(5x\) and \(3x\) are both terms with \(x.\)
  • The terms \({n}^{2}\) and \(9{n}^{2}\) both have \({n}^{2}.\)

Terms are called like terms if they have the same variables and exponents. All constant terms are also like terms. So among the terms \(5x,7,{n}^{2},4,3x,9{n}^{2},\)

\(7\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}4\phantom{\rule{0.2em}{0ex}}\text{are like terms.}\)

\(5x\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}3x\phantom{\rule{0.2em}{0ex}}\text{are like terms.}\)

\({n}^{2}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}9{n}^{2}\phantom{\rule{0.2em}{0ex}}\text{are like terms.}\)

Like Terms

Terms that are either constants or have the same variables with the same exponents are like terms.

Example

Identify the like terms:

  1. \(\phantom{\rule{0.2em}{0ex}}{y}^{3},7{x}^{2},14,23,4{y}^{3},9x,5{x}^{2}\)
  2. \(\phantom{\rule{0.2em}{0ex}}4{x}^{2}+2x+5{x}^{2}+6x+40x+8xy\)

Solution

\(\phantom{\rule{0.2em}{0ex}}{y}^{3},7{x}^{2},14,23,4{y}^{3},9x,5{x}^{2}\)

Look at the variables and exponents. The expression contains \({y}^{3},{x}^{2},x,\) and constants.

The terms \({y}^{3}\) and \(4{y}^{3}\) are like terms because they both have \({y}^{3}.\)

The terms \(7{x}^{2}\) and \(5{x}^{2}\) are like terms because they both have \({x}^{2}.\)

The terms \(14\) and \(23\) are like terms because they are both constants.

The term \(9x\) does not have any like terms in this list since no other terms have the variable \(x\) raised to the power of \(1.\)

\(\phantom{\rule{0.2em}{0ex}}4{x}^{2}+2x+5{x}^{2}+6x+40x+8xy\)

Look at the variables and exponents. The expression contains the terms \(4{x}^{2},2x,5{x}^{2},6x,40x,\text{and}\phantom{\rule{0.2em}{0ex}}8xy\)

The terms \(4{x}^{2}\) and \(5{x}^{2}\) are like terms because they both have \({x}^{2}.\)

The terms \(2x,6x,\text{and}\phantom{\rule{0.2em}{0ex}}40x\) are like terms because they all have \(x.\)

The term \(8xy\) has no like terms in the given expression because no other terms contain the two variables \(xy.\)


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