# Mathematics Knowledge Study Guide for the ASVAB

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## General Information

The Mathematics Knowledge section of the ASVAB exam measures your knowledge of various math areas, mainly related to algebra and geometry. The CAT-ASVAB has 16 questions to be answered in 20 minutes; the paper-and-pencil version has 25 questions in 24 minutes. Questions in this ASVAB section can be stated simply as math problems or presented in a word problem format.

While this section does not focus specifically on arithmetic skills, they are still required to tackle the algebraic and geometrical questions you will see. So, before using this study guide, it would be a good idea to review our study guide for the Arithmetic Reasoning section to be sure you have a good handle on those basic concepts. You will need them.

## Algebra: Part 1

Whereas arithmetic deals with mathematical operations with numbers, algebra introduces the idea of using letters to stand for numbers, calling them variables. Algebra gives us rules for handling combinations of numbers and variables that can be much more complex than are seen in arithmetic.

For a simple example, in arithmetic, you might see something like $$2 \times \_\_\_ = 18$$ and be asked what number goes in the blank. In algebra, the same problem would be presented as the equation $$2x=18$$, where $$x$$ is a variable that stands for a number, and $$2$$ is referred to as a coefficient. You would be asked for the value of $$x$$ that would make the equation true.

### Algebraic Expressions

In algebra, numbers like $$7$$ or $$93$$ are known as constants (because they’re not variables). An algebraic expression is any combination of constants, variables, and math operation symbols, but it does not include an equal sign. If there is an equal or inequality sign, then it is an equation, not an expression.

A few examples of expressions:

$32t$ $5x \cdot x +8$ $(x+3)(x-2)$ $4y^2-3y-9$

#### Terms

Terms are the individual parts that make up an algebraic expression, consisting of a constant or a constant times a variable raised to some power and separated by a plus or minus sign. The expression $$r^2 -5r +6$$ has three terms: $$r^2$$, $$-5r$$, and $$6$$.

##### Combining Terms

Expressions can sometimes be simplified by combining two or more terms into one. Terms that are constants or that have the same variable with the same exponent (called like terms) can be combined.

These are examples of like terms:

$$8x$$ and $$5x$$

$$9y^3$$, $$y^3$$, and $$11y^3$$,

Here is an example of combining terms:

$5x^2 +8x -2x^2 +9 +2x +x^2$

First, group like terms together:

$5x^2 -2x^2 +x^2 +8x+2x+9$

Now, combine the like terms:

$4x^2+10x+9$

When adding or subtracting terms that are constants, simply add or subtract as you would in an arithmetic problem.

When adding or subtracting terms with variables, be sure they are like variables and just add or subtract the coefficients and write the variable after the coefficient.

For example: $$12x^2 + 3x^2$$

Add $$12 +3$$ and get $$15$$. Now, write $$x^2$$ after the $$15$$ to get $$15x^2$$.

##### Multiplying and Dividing Terms

Expressions may be written showing terms multiplied or divided by each other. Unlike adding and subtracting, terms don’t need to be like terms to multiply or divide them. Here are a couple of examples.

To do the calculation $$9x^2y \cdot 3x^3y^3$$, multiply the constants first, then multiply the variables, using the laws of exponents if needed. Then put them together as one term.

Example 1

$$9 \cdot 3$$ and $$x^2y \cdot x^3y^3$$

$27x^5y^4$

Example 2

$\dfrac{30a^4bc^2}{6abc}$ $\dfrac{30}{6} \cdot \dfrac{a^4bc^2}{abc}$ $5a^3c$

#### Types of Expressions

Expressions can be classified according to how many terms they have.

##### Monomials

Monomials have only one term, such as:

$2x^3$ $5xy^2z$
##### Binomials

Binomials have two terms, such as:

$x^2-y^2$ $9x-4$
##### Polynomials

Polynomials are algebraic expressions consisting of several terms, such as constants, variables, and exponents, combined together by addition, subtraction, multiplication, and division, excluding division by variables. Examples are:

$2x^2-3x+5$ $x^3 +3x^2 -5x +8$

#### Simplifying and Evaluating Expressions

Many times the expressions you run across can be simplified, meaning to make the expression have fewer or smaller terms. A common way to do this is to combine like terms, as was discussed above. Sometimes multiplying two binomials will produce a trinomial, which is simpler, having only three terms instead of four. Below is an example of using both techniques.

$(2x+3)(x-2)$

Using the FOIL method, described later in this guide, we get:

$2x^2-4x+3x-6$

Combining $$-4x+3x=-x$$ gives us this trinomial:

$2x^2-x-6$
##### Distributing

The distributive property comes up a lot in algebra. It involves multiplying a monomial times a binomial or higher.

The general pattern is $$a(b+c) = ab+ac$$.

Here is an example:

$5y(y^2-9)$ $5y \cdot y^2 + 5y \cdot (-9)$ $5y^3-45y$
##### Factoring

This is very useful in solving various problems involving polynomials.

Factor the polynomial:

$x^2 – 7x + 12$ $(x +\;?\;)(x +\;?\;)$

Start with the general format for factoring a polynomial in the second degree. Think of two numbers which, when multiplied, will result in +12, and when added together will result in –7. That’s quite easy:

$(–3) \times (–4) = +12 \text{ and }(–3)+(–4) = –7$

Plug –3 and –4 into the general format:

$(x – 4)(x – 3)$

These are the factors of the polynomial.

##### Solving Polynomial Equations

When a problem tells you to solve the polynomial equation, it is the same as telling you to find the roots of the equation, or solve for the values for x that will make the equation true. Using the same illustration as above and continuing to find the roots, set each of the factors equal to zero and solve for x:

$$x^2 – 7x + 12$$=0

$$x – 4 = 0$$ and $$x – 3 = 0$$

$$x = 4$$ and $$x = 3$$

These are the roots of the polynomial, or the values of x.

Note: Be very careful to note whether you are being asked to find the factors or the roots. The answers will not be the same.

##### The FOIL Method

This is a method for multiplying two binomials that helps us get the terms multiplied in the right combinations. FOIL stands for First, Outside, Inside, Last, which itself is short for first pair, outside pair, inside pair, and last pair. Here’s how it works. Start with the binomials below:

$(2x+3)(5x+1)$

Multiply the First pair of numbers: $$2x \cdot 5x=10x^2$$
Multiply the Outside pair of numbers: $$2x \cdot 1=2x$$
Multiply the Inside pair of numbers: $$3 \cdot 5x=15x$$
Multiply the Last pair of numbers: $$3 \cdot 1=3$$
Add the four results, combining the $$2x$$ and the 15x and get

$10x^2 + 17x +3$

A quadratic expression is any expression in which the highest power of the variable is $$2$$.

$x^2+9x-11\quad\quad \quad 3x^2 -99 \quad\quad\quad 3y +10y -7y^2$

Factoring quadratic equations is the same as factoring quadratic expressions that we saw above. The only difference is that there is an equal sign and a number to the right, often zero. This is almost always done as a step in solving the equation. See the example below.

Factor this:

$x^2-4x-32=0$ $(x + \text{?})(x + \text{?})$

What two numbers add up to $$-4$$ and have a product of $$-32$$?

The answer is $$4$$ and $$-8$$, so we have these factors.

$(x+4)(x-8)$
##### Three Types of Quadratic Equations

There are three patterns that show up often in quadratic equations. If you can recognize them on sight, you can save some factoring or FOIL work.

This one is called the difference of two squares because that’s exactly what the product is. Notice that there is no middle term.

$(x+a)(x-a) = x^2-a^2 \quad\quad \text{ Example } (x+5)(x-5) = x^2-25$

These two are called perfect squares.

$(x+a)^2 = x^2+2ax+a^2\quad\quad \text{ Example } (x+8)^2 = x^2 +16x +64$ $(x-a)^2 = x^2-2ax+a^2 \quad\quad\text{ Example } (x-3)^2 = x^2 -6x +9$

These are probably most useful when you are given the equation and are trying to factor it.

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