GRE Algebra - Expressions

Most algebra questions on the GRE will present you with an expression in one form or another. Now, what does that mean?

An expression is a relationship between several numbers or variables or a combination of both. The most common algebraic expressions involve the variables x and y, but a variable can be given any name, and can be designated by an alphabet character or by a non alphabet character. For example

3x+4y = 3
3p+4m= 3
3#+4& = 3

are all equivalent expressions, because when x, p and # are all equal to 1 and y, m and & are all 0, then each of the equations is true, so each of the three expressions actually produces the same relationship. Note that this relationship works only if x, p and # are all the same values, and y, m, and & are all the same values. Yes, it's that simple.

There are several rules that you should be familiar with when manipulating expressions, and these rules follow the rules of arithmetic. So again, here is an example of why you need to have a good understanding of the basics before you can begin to understand more complicated topics that might appear on the GRE exam.

The Associative Property enables grouping of numbers and variables. Just like you can do (2 + 4) + 3 = 2 + (4 +3), you can also do (x + y) + z = x + (y + z). So the associate property simply allows you to regroup different parts of an expression.

The Distributive Property is the equivalent of the associate property, except for multiplication. So just like you can do 3(4 + 5) = 3(4) + 3(5), you can do x(y+z) = (xy)+(xz). The distributive property simply allows you to "multiply out".

There are several rules for squares, namely:

(x+y)(x-y) = x2-y2
(x+y)(x+y) = x2+2xy+y2

Again, as in the previous cases, the properties for the squares of variables are identical to what you would do if you were doing the same operations using numbers:

(1+2)(1-2) = (3)(-1) = -3 = 12 - 22 = 1-4 = -3.

Do a few of these to convince yourself and then you'll see that algebra is not very complicated. Also another property of squares that you should be familiar with is binomial squares, or squares of 2 numbers. For example:

(x+y)2 = x2 + 2xy + y2
(x-y)2 = x2 - 2xy + y2

Again these properties follow from the answers that you would get if you were doing the same operations on numbers rather than variables. Try it out:

(1+2)2 = 12+2(1)(2) + 22 = 1 + 4 + 4 = 9.