An algebraic expression is a collection of constants and variables that are combined using one or more of the four fundamental operations namely, addition, subtraction, multiplication and division (except division by zero).
The essence of algebra lies in representing quantities as symbols other than numerals. This is the advantage of applying algebra and not arithmetic alone in solving practical problems. These different symbols are grouped into expressions, which in turn bring meaning to equations and inequalities.
Every symbol that has no fixed value and stands for a number is called a placeholder symbol. In arithmetic, students meet problems like 4 + 3 = ç, 3 x 2 = â, 12 ÷ ã = 3, 2n = 8. Algebra does not use ã, ç, and â as symbols but uses x, n, y, or any letter to represent numbers. A letter that is used as a placeholder symbol that has no fixed value is called a variable while a symbol that has a fixed value is called a constant.
A mathematical phrase that contains a variable is an open phrase. A number phrase is an expression that does not contain a variable. It is also referred to as a numerical expression. The English phrase ‘a certain number added to 5’ is translated to an open phrase ‘n + 5’ where n stands for a certain number. The English phrase ‘seven added to 5’ is translated to a number phrase ‘7 + 5’.
Expressions like 8 + 2, 12 – 2, 5 x 2, and 20 ÷ 2 are some number phrases for the number 10. Expressions like , - 2a, 2n + 8, 2(l + w) are examples of open phrases. Another name for open phrase is algebraic expression. An expression composed of constants, variables, grouping symbols, and operation symbols, is called an algebraic expression.
In an open phrase “-7x + 12”, -7x and 12 are the terms of the expression. In the term -7x, -7 is the constant, also called the numerical coefficient of x while x is the variable, or the literal coefficient of -7. The numerical coefficient of a term is written before the literal coefficient. A term is an indicated product or quotient of coefficients. The term 12 is the constant term, which does not have any indicated literal coefficient. Terms in an algebraic expression are separated by the plus (+) or minus (-) sign.