The use of variables to represent numbers is what differentiates algebra from arithmetic. Calculations in algebra may involve solving for a value of a variable that makes an equation true, or they may involve substituting different values for a variable in an expression. On the GRE, you will see some questions that are strictly algebra based, but you will also see questions that involve the use of algebra along with reasoning, problem solving, and data interpretation skills. For those reasons, algebra is an important area on which to focus your review. You must understand basic equations and how to solve them. A good place to start is with a review of algebraic terminology to ensure that you understand directions, questions, and explanations as you go along.
Variable: A letter used to represent a quantity whose value is unknown.
Examples: The letters x, y, n, a, b, and c are used frequently to represent variables.
Term: A term is a numerical constant or the product (or quotient) of a numerical constant and one or more variables.
Examples: 3x, 4x², and 2ac
Expression: An algebraic expression is a combination of one or more terms. Terms in an expression are separated by either addition or subtraction signs.
Examples: 3xy, 4ab − 5cd, and x²+ x − 1.
Coefficient: In the term 3xy, the multiplier 3 is called a coefficient. In a simple term such as z, 1 is the coefficient.
Constant: A value that does not change.
Example: In the expression x + 7, the number 7 is a constant.
Monomial: A single term, such as −6x or 2a².
Polynomial: The general name for expressions with more than one term.
Binomial: A polynomial with exactly two terms.
Trinomial: A polynomial with exactly three terms.
So what are the common ways your algebra skills will be tested? Here’s what you can expect.
Basic Algebra Operations
Combining Like Terms
The process of simplifying an expression by adding together or subtracting term sthat have the same variable factors is called combining like terms.
Adding and Subtracting Polynomials
To add or subtract polynomials, combine like terms.
Factoring Algebraic Expressions
Factoring a polynomial means expressing it as a product of two or more simpler expressions. Common factors can be factored out by using the distributive law.
Advanced Algebra Operations
Substitution, a process of plugging values into equations, is used to evaluate an algebraic expression or to express it in terms of other variables. Replace every variable in the expression with the number or quantity you are told is its equivalent. Then carry out the designated operations, remembering to followthe order of operations (PEMDAS).
When you manipulate any equation, always do the same thing on both sides of the equal sign. Otherwise, the two sides of the equation will no longer be equal. To solve an algebraic equation without exponents for a particular variable, you have to manipulate the equation until that variable is on one side of the equal sign with all numbers or other variables on the other side. You can perform addition, subtraction, or multiplication; you can also perform division, as long as the quantity by which you are dividing does not equal zero.
Solving for One Unknown in Terms of Another
In general, in order to solve for the value of an unknown, you need as many distinct equations as you have variables. If there are two variables, for instance, you need two distinct equations. However, some GRE problems do not require you to solve for the numerical value of an unknown. Instead, you are asked to solve for one variable in terms of the other(s). To do so, isolate the desired variable on one side of the equation and move all the constants and other variables to the other side.
We’ve already discovered that you need as many different equations as you have variables to solve for the actual value of a variable. When a single equation contains more than one variable, you can only solve for one variable in terms of the others. This has important implications for Quantitative Comparisons. To have enough informationto compare the two quantities, you usually must have at least as many equations as you have variables. On the GRE, you will often have to solve two simultaneous equations, that is, equations that give you different information about the same two variables. There are two methods for solving simultaneous equations, one involving substitution and the other requiring you to add multiple equations together.
Don’t panic if you see strange symbols like ★, ✧, and ♦ in a GRE problem. Problems of this type usually require nothing more than substitution. Read the question stem carefully for a definition of the symbols and for any examples of how to use them. Then, just follow the given model, substituting the numbers that are in the question stem.
Sequences are lists of numbers. The value of a number in a sequence is related toits position in the list. Sequences are often represented on the GRE as follows:
s s₂, s₃,…sn,…
The subscript part of each number gives you the position of each element in theseries. s1 is the first number in the list, s2 is the second number in the list, and so on. You will be given a formula that defines each element. For example, if you are told that sn = 2n + 1, then the sequence would be (2 × 1) + 1, (2 × 2) + 1, (2 × 3) + 1,…,or 3, 5, 7,…