Algebraic expressions are combinations of variables , numbers, and at least one arithmetic operation.

For example, 2 x + 4 y − 9 is an algebraic expression.

** Term: ** Each expression is made up of terms. A term can be a signed number, a variable, or a constant multiplied by a variable or variables.

** Factor: ** Something which is multiplied by something else. A factor can be a number, variable, term, or a longer expression. For example, the expression 7 x ( y + 3 ) has three factors: 7 , x , and ( y + 3 ) .

** Coefficient: ** The numerical factor of a multiplication expression that contains a variable. Consider the expression in the figure above, 2 x + 4 y − 9 . In the first term, 2 x , the coefficient is 2 : in the second term, 4 y , the coefficient is 4 .

** Constant: ** A number that cannot change its value. In the expression 2 x + 4 y − 9 , the term 9 is a constant.

** Like Terms: ** Terms that contain the same variables such as 2 m , 6 m or 3 x y and 7 x y . If an expression has more than one constant terms, those are also like terms.

Expression |
Word Phrases |

n + 5 |
Sum of a number and 5 |

m − 7 |
Difference of a number and 7 |

6 x |
Product of 6 and a number |

y ÷ 9 |
Quotient of a number and 9 |

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** Example: **

Identify the terms, like terms, coefficients, and constants in the expression.

9 m − 5 n + 2 + m − 7

First, we can rewrite the subtractions as additions.

9 m − 5 n + 2 + m − 7 = 9 m + ( − 5 n ) + 2 + m + ( − 7 )

So, the ** terms ** are 9 m , ( − 5 n ) , m , 2 , and ( − 7 ) .

** Like terms ** are terms that contain the same variables.

9 m and 9 m are a pair of ** like terms ** . The constant terms 2 and − 7 are also like terms.

** Coefficients ** are the numerical parts of a term that contains a variable.

So, here the ** coefficients ** are 9 , ( − 5 ) , and 1 . ( 1 is the coefficient of the term m .)

The ** constant ** terms are the terms with no variables, in this case 2 and − 7 .

Algebraic expressions must be written and interpreted carefully. The algebraic expression 5 ( x + 9 ) is ** not ** equivalent to the algebraic expression, 5 x + 9 .

See the difference between the two expressions in the table below.

Word Phrases |
Algebraic Expression |

Five times the sum of a number and nine | 5 ( x + 9 ) |

Nine more than five times a number | 5 x + 9
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In writing expressions for unknown quantities, we often use standard formulas. For example, the algebraic expression for “the distance if the rate is 50 miles per hour and the time is T hours” is D = 50 T (using the formula D = R T ).

An expression like x n is called a power. Here x is the base, and n is the exponent. The exponent is the number of times the base is used as a factor. The word phrase for this expression is ” x to the n th power.”

Here are some of the examples of using exponents.

Word Phrases |
Algebraic Expression |

Seven times m to the fourth power | 7 m 4 |

The sum of x squared and 12 times of y | x 2 + 12 y |

x cubed times y to the sixth power |
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