Algebra is just a form of arithmetic in which symbols (usually letters) stand for numbers. You use algebra to solve equations and to find the value of a variable. 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 A has 5 more pencils than B, then the number of pencils that A has is B+5.
Although the GMAT doesn’t specifically test you on the definitions of variable, constant, and coefficient, it does expect you to know these concepts when they come up in the questions.
You will see a lot of variables in algebra problems. They are the symbols that stand for numbers. Usually the symbols take the form of letters and represent specific numeric values. The value of variables can change depending on the equation they are in. For example, if a store charges different prices for apples and oranges and you buy two apples and four oranges, so to express the transaction in algebraic terms, you use variables to stand in for the price of apples and oranges, something like 2a and 4o.
In contrast, constants are numbers with values that don’t change in a specific problem. Letters may also be used to refer to constants, but they don’t change their value in an equation as variables do.
Single constants and variables or constants and variables grouped together form terms. Terms are any set of variables or constants you can multiply or divide to form a single unit in an equation. You can combine these single parts in an equation that applies addition or subtraction. For example, the following algebraic expression has three terms: ax2 + bx + c. The first term is ax2, the second term is bx, and the third term is c.
Terms often form expressions. An algebraic expression is a collection of terms that are combined by addition or subtraction and are often grouped by parentheses, such as (x + 2), (x – 3c), and (2x –3y).
A coefficient is a number or symbol that serves as a measure of a property or characteristic. In 2a + 4o, the variables are a and o, and the numbers 2 and 4 are the coefficients of the variables. This means that the coefficient of the variable a is 2 and the coefficient of the variable o is 4.
In an algebraic expression, terms involving the same variable, even if they have different coefficients, are called like terms. For example, in the expression 3x + 4y – 2x + y, 3x and –2x are like terms because they both contain the single x variable; 4y and y are also like terms because they both contain the y variable and only the y variable. You can combine (add or subtract) like terms together, but you can’t combine unlike terms.
Types of Expressions
Expressions carry particular names depending on how many terms they contain.
A monomial is an expression that contains only one term, such as 4x or ax2. Therefore, a monomial is also referred to as a term in an algebraic expression.
A polynomial (Poly means many) is an expression that has more than one term. These multiple terms can be added together or subtracted from one another. For example, a2 - b2, or ab2 + 2ab + c. Polynomials can be more specific, depending on how many terms they contain. For example, a binomial is a specific kind of polynomial, that contains two terms, such as a + b or 2a + 3. A trinomial is a polynomial with three terms, like 4x2 + 3y - 8.
A famous trinomial that you should be very familiar with for the GMAT is the expression known as a quadratic polynomial, which is ax2 + bx + c.
Simplifying Algebraic Equations
Symbols like +, –, ×, and ÷ are common to arithmetic and algebra. They symbolise the operations you perform on numbers. Arithmetic uses numbers with known values, such as 5 + 7 = 12, in its operations, but algebraic operations deal with unknowns, like x + y = z. This algebraic equation can’t produce an exact numerical value because you don’t know what x, y and z represent.
When working with algebraic expressions, it is necessary to simplify them by factoring or combining like terms. For example, the expression 6x + 5x is equivalent to (6+5)x, or 11x. In the expression 9x - 3y, 3 is a factor common to both the terms: 9x - 3y = 3(3x-y). In the expression 5x2 + 6y, 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.
To multiply two algebraic expressions, each term of one expression is multiplied by each term of the other expression. For example:
(3x - 4)(9y + x) = 3x(9y + x) - 4(9y + x) = (3x)(9y) + (3x)(x) + (-4)(9y) + (-4)(x) = 27xy + 3x2 - 36y - 4x
An algebraic expression can be evaluated by substituting values of the unknowns in the expression. For example, if x = 3 and y = -2, then 3xy - x2 + y can be evaluated as 3(3)(-2) - (3)2 + (-2) = -18 - 9 - 2 = -29.
FOIL Method for Multiplying Binomials
You can multiply binomials by using the FOIL method. FOIL is an acronym for first, outer, inner, last, which indicates the order that you multiply the terms from one binomial by the terms of the second binomial before adding their products. For example,
(4x - 5)(3x + 8)
Multiply the first terms in each binomial: 4x and 3x.
4x × 3x = 12x2
Then, multiply the outer terms (4x and 8) to get 32x and the inner terms (3x and –5) to get –15x. You can add the products at this point because they are like terms.
32x - 15x = 17x
Last, multiply the last terms.
-5 × 8 = -40
Combine the products to form the resulting expression. So,
(4x - 5)(3x + 8) = 12x2 + 17x - 40
Formulas for Algebra
To save time on the GMAT, use the following factors and their resulting equations:
(a + b)2 = a2 + 2ab + b2
(a - b)2 = a2 - 2ab + b2
Factors are the numbers you multiply together to get a product. So, factoring a value means you write that value as a product of its factors. For the GMAT, you should know how to pull out the common factors in expressions and the two binomial factors in a quadratic polynomial.
To simplify polynomials for complex problems, extract their common factors by dividing each term by the factors that are common to every term. For example, to find the common factors of the terms in the expression - 14x3 - 35x6, follow these steps:
1. Consider the coefficients
Because –7 is common to both –14 and –35, take this factor out of the expression by dividing both terms by –7. Then put the remaining expression in parentheses next to the common factor: -7(2x3 + 5x6)
2. Look at the variables
Because x3 or a multiple of it is common to both terms, divide both terms in parentheses by x3, multiply x3 by the other common factor (–7), and put the remaining expression in parentheses.
So, - 14x3 - 35x6 = -7x3(2 + 5x3)