5 Algebraic Expressions and Simplification
Learn how variables, terms, and algebraic expressions represent quantities, and how to simplify expressions by combining like terms and applying the distributive property.
Variables, terms, and expressions
A is a letter that represents a number, such as or . An combines numbers, variables, and operation symbols. For example, is an expression because it has no equals sign.
A is a number, a , or a product of numbers and variables. In the , is the of ; in the expression , is a .
Expressions can describe quantities. If a notebook costs dollars, then represents the cost of three notebooks, and represents that cost plus a -dollar fee.
Recognizing and combining
have the same variables raised to the same powers, though their coefficients may differ. For example, and are , as are and . Constants are with other constants.
Terms such as and , or and , are unlike terms. To combine , add or subtract their coefficients while keeping the part unchanged:
Unlike terms cannot be combined by addition or subtraction. For example, is already simplified because its and are unlike terms.
The
The says that multiplying a sum by a factor is equivalent to multiplying every in the sum by that factor:
For example, . The outside factor applies to every inside the parentheses. This also applies when the factor is negative:
The signs matter: . The property also works in reverse; for example, .
A method for simplifying expressions
To simplify an expression, distribute to remove parentheses when needed, group , combine their coefficients while tracking their signs, and check that no remain.
For example, simplify :
A common mistake is distributing to only the first : , not . Another is combining unlike terms. The result cannot be reduced further because the - and the are different kinds of terms.