GMAT Quantitative Review

3.0 Math Review

3.2 Algebra

Algebra is based on the operations of arithmetic and on the concept of an unknown quantity, or variable. Letters such as x or n are used to represent unknown quantities. For example, suppose Pam has 5 more pencils than Fred. If F represents the number of pencils that Fred has, then the number of pencils that Pam has is . As another example, if Jim”s present salary S is increased by 7%, then his new salary is 1.07S. A combination of letters and arithmetic operations, such as , and , is called an algebraic expression.

The expression consists of the terms 19x², −6x, and 3, where 19 is the coefficient of x², −6 is the coefficient of x¹, and 3 is a constant term (or coefficient of ). Such an expression is called a second degree (or quadratic) polynomial in x since the highest power of x is 2. The expression is a first degree (or linear) polynomial in F since the highest power of F is 1. The expression

is not a polynomial because it is not a sum of terms that are each powers of x multiplied by coefficients.

1. Simplifying Algebraic Expressions

Often when working with algebraic expressions, it is necessary to simplify them by factoring or combining like terms. For example, the expression is equivalent to , or 11x. In the expression , 3 is a factor common to both terms: . In the expression , there are no like terms and no common factors.

If there are common factors in the numerator and denominator of an expression, they can be divided out, provided that they are not equal to zero.

For example, if , then is equal to 1; therefore,

To multiply two algebraic expressions, each term of one expression is multiplied by each term of the other expression. For example:

An algebraic expression can be evaluated by substituting values of the unknowns in the expression. For example, if and , then can be evaluated as