Algebra

(x/y)^r

(x^r)/(y^r)

(x^r)(y^r)

(xy)^r

x^0

1

x^(−r)

1/(x^r)

Factoring: The Distributive Property in Reverse

A(B + C) = AB + AC ∴ AB + AC = A(B + C)

Eliminate Variables (Elimination) e.g. x+y=7 & x-y=1

Add or subtract both sides of two or more equations to solve the equation directly. This is permissible because both sides are equal.

Harder Factoring: What is the greatest prime factor of 12!*11! + 11!*10!? (A) 2 (B) 7 (C) 11 (D) 19 (E) 23

First factor out 11! 11!(12!+10!) Now factor out 10! 11!*10!(11*12*1) Simplify 11!*10!(133) Test for primes in 133: (7*19) Since 19 is not a factor of 11! or 10! answer D is correct.

FOIL

First, Outside, Inside, Last ax²+bx+c=0 Factor: (x+y)(x+z)=0 x*x+x*z+x*y+y*z

Eliminate Variables (Substitution) e.g. x+y=6 & x=2y e.g.2

If x + y = 6 and x = 2y then 2y+y = 6 → 3y = 6 → y = 2 ∴ x + 2 = 6 → x = 4

Multiple Inequalities

It is easiest to treat each inequality as a different statement for the purposes of performing algebraic operations e.g. 2 < x + 2 < 4 Becomes 2<x+2 and x+2<4 Then, you can take these conclusions and reform the brackets: 0 < x < 2

Multiplying two like variables

Multiplying two like variables results in a single variable raised to the power of the sum of the powers of the variables being multiplied.

Multiply by 1 e.g. (4a/3)+a

On the GMAT many expressions and equations that you encounter will involve fractions and variables. You can often simplify them by multiplying by 1.

Inequalities

Solve like a normal algebraic equation. However, if multiplying both sides by a negative number the inequality is reversed.

Multiple Variables and Special Cases

Solving for multiple variables will most likely require you to have as many equations as variables. There are a few exceptions: • Definitions: If one of the variables is constrained by a definition such as "x is a positive integer" or "y is a prime number," the pool of possible solutions may be narrowed to a point where you can solve with fewer equations. • Inequalities: Inequalities, such as x < 5, can also limit the pool of possibilities and allow you to solve for multiple variables with fewer solutions. A good rule of thumb is that, when faced with more variables than equations, you should: 1. Search for more equations, or 2. Reduce the number of variables

Factoring Quadratics

The reverse of the FOIL method

Addition and Subtraction (algebra

Two like variables of different powers, such as x and x². Neither the expression x + x² nor the expression x - x² can be condensed into one term (although they could be factored).

Standard quadratic equation

ax²+bx+c=0 e.g. x²+6x+5 = 0 (factor by grouping): a+b = 6 & a*b = 5 therefore, a = 1 and b = 5 x²+5x+x+5 = 0 therefore x(x+5)*1(x+5) = 0 (x+1)(x+5) = 0 So x = -1 or x = -5

Combine Like Terms/Factor e.g. What is the value of x + y? 2x + 3y + 5x +4y = 21 e.g. 2 4⁸ + 4⁸ + 4⁸ + 4⁸ = 4x. What is x? (A) 4 (B) 8 (C) 9 (D) 16 (E) 32

e.g. 1 2x+5x+3y+4y = 21 7x + 7y = 21 7(x+y) = 21 x+y = 3 e.g. 2 o start the problem, you must simplify the left side of the equation: 4⁸ + 4⁸ + 4⁸ + 4⁸ = 4(4⁸). Now that you have multiplication 4(4⁸) you can then add exponents to see that expression equals 49. If 49 = 4x, then x = 9. Answer choice C is correct.

Factoring (Finding roots) e.g. (x²)-2x-35 = 0

first find 2 numbers whose sum is equal to -2 and whose product is equal to -35 → a+b = -2 & a*b = -35 (x+5)(x-7) = 0 a = 5 & b = -7

Quadratics and Multiple Solutions e.g. x²=81

x is either 9 or -9, as each value would satisfy the equation.

Functions

x is the input and f(x) is the output The domain of a function is the set of allowable inputs

(x^r )(x^s)

x^(r+s)

(x^r)/(x^s)

x^(r-s)

(x^r)^s

x^(r∗s)

Absolute Value e.g. |x + 5| = 10

|x + 5| = 10 x+5=10 or -1(x+5)=10 x = 5 or -x-5=10 x = 5 or x = -15