Practice questions for this study guide can be found at:
Algebra and Probability Practice Problems for McGraw-Hill's ASVAB
Monomials, Binomials, and Polynomials
You can guess by the prefix mono- that a monomial has something to do with "one." A monomial is a mathematical expression consisting of only one term. Examples include 12x, 3a2, and 9abc.
- A binomial (the prefix bi- means "two") has exactly two terms: 12z + j.
- A polynomial, as indicated by the prefix poly-, meaning "many," has two or more terms.
Examples include x + y, x + y + z, and y2 – 2z + 12.
Multiplying Monomials When multiplying monomials, multiply any numbers, then multiply unknowns. Add any exponents. Keep in mind that in a term like x or 2x, the x is understood to have the exponent 1 even though the 1 is not shown.
Dividing Monomials To divide monomials, divide the numbers and subtract any exponents (the exponent of the divisor from the exponent of the number being divided).
Adding and Subtracting Polynomials Arrange the expressions in columns with like terms in the same column. Add or subtract like terms.
Multiplying Polynomials To multiply polynomials, multiply each term in the first polynomial by each term in the second polynomial. The process is just like regular multiplication. For example, if you multiply 43 times 12, the problem looks like this:
Dividing a Polynomial by a Monomial Just divide the monomial into each term of the polynomial.
Dividing a Polynomial by a Polynomial To divide a polynomial by another polynomial, first make sure the terms in each polynomial are in descending order (i.e., cube → square → first power).
For example, 6c + 3c2 + 9 should be written 3c2 + 6c + 9.
10 + 2c + 5c2 should be written 5c2 + 2c + 10.
Then use long division to solve the problem.
Factoring a Polynomial A factor is a number that is multiplied to get a product. Factoring a mathematical expression is the process of finding out which numbers, when multiplied together, produce the expression.
To factor a polynomial, follow these two steps:
- Find the largest common monomial in the polynomial. This is the first factor.
- Divide the polynomial by that monomial. The result will be the second factor.
Special Case: Factoring the Difference between Two Squares
Examples
Factor the following expression:
y2 – 100
In this expression, each term is a perfect square; that is, each one has a real-number square root. The square root of y2 is y, and the square root of 100 is 10.
When an expression is the difference between two squares, its factors are the sum of the squares (y +10) and the difference of the squares (y – 10). Multiplying the plus sign and the minus sign in the factors gives the minus sign in the original expression.
Factoring Polynomials in the Form ax2 + em>bx + c, where a, b, and c are numbers Remember that you want to find two factors that when multiplied together produce the original expression.
Examples
Factor the expression:
First, you know that x times x will give x2, so it is likely that each factor is going to start with x.
Now you need to find two factors of 6 that when added together give the middle term of 5. Some options are 1 and 6 and 2 and 3. 2 and 3 add to 5, so add those numbers to your factors. Now you have
Finally, deal with the sign. Since the original expression is all positive, both signs in the factors must be positive. So the two factors must be
Check your work by multiplying the two factors to see if you come up with the original expression.
(x + 2)(x + 3) = x2 + 5x + 6
Factor the expression:
6x2 can be factored into either (6x)(x) or (2x)(3x). Using the latter, the first terms in our factors are as follows:
Now let's consider the –8. Factors of 8 can be (8)(1) or (2)(4). Let's try 2 and 4, so our factors are now:
Now for the signs. In order to get a minus 8 in the original expression, one of the numbers must be a negative and the other a positive. Let's try making the 4 negative, making the factors:
(2x + 2)(3x – 4) = 6x2 – 2x – 8
That's close, but the original expression was 6x2 + 8x – 8, not 6x2 – 2x – 8. What if we switched the numbers 2 and 4?
(2x + 4)(3x – 2) = 6x2 + 8x – 8
Now the factors give the original expression when multiplied together, so this is the correct answer.
Solving Quadratic Equations
A quadratic equation is one that is written in the form ax2 + bx + c, where a, b, and c are numbers.
To solve an equation in this form for x, set the expression equal to zero. Note that x will have more than one value.
Follow these steps:
- Put all the terms of the expression on one side of the = sign and set it equal to zero.
- Factor the equation.
- Set each factor equal to zero.
- Solve the equations.
Algebraic Fractions
An algebraic fraction is a fraction containing one or more unknowns.
Reducing Algebraic Fractions to Lowest Terms
To reduce an algebraic fraction to lowest terms, factor the numerator and the denominator. Cancel out or divide common factors.
Example
Reduce to lowest terms:
Example
Reduce to lowest terms:
Adding or Subtracting Algebraic Fractions with a Common Denominator To add or subtract algebraic functions that have a common denominator, combine the numerators and keep the result over the denominator. Reduce to lowest terms.
Examples
Adding or Subtracting Algebraic Fractions with Different Denominators To add or subtract algebraic fractions that have different denominators, examine the denominators and find the least common denominator. Then change each fraction to the equivalent fraction with that least common denominator. Combine the numerators as shown in the previous section. Reduce the result to lowest terms.
Examples
Note that abis the least common denominator.
In this example, 12xis the least common denominator, as 4xand 6xboth divide into it.
Note that you have to multiply in order to make each term contain the least common denominator.
Multiplying Algebraic Fractions When multiplying algebraic fractions, factor any numerator and denominator polynomials. Divide out common terms where possible. Multiply the remaining terms in the numerator and denominator together. Be sure that the result is in lowest terms.
Examples
Multiply the following fractions:
Divide out common terms, then multiply. Reduce to lowest terms.
Multiply the following fractions:
Dividing Algebraic Fractions To divide algebraic fractions, follow the same process used to divide regular fractions: invert one fraction and multiply.
Examples
Divide the following fractions:
Divide the following fractions:
Graphing on a Number Line
You can represent a number as a point on a number line, as shown in the following examples. Representing a number on a number line is called graphing. Note that whole numbers on the line are equally spaced. Note too that in these examples, both positive and negative numbers are represented.
On the number line, positive numbers are shown to the right of zero. Negative numbers are shown to the left of zero. The positive number +3 is three units to the right of zero. The negative number –2 is two units to the left of zero.
Graphing on a Coordinate Plane
A coordinate plane is based on an xaxis (horizontal number line) and a yaxis (vertical number line). The axes intersect at their zero points. This point of intersection is called the origin. Every point on the plane has both an xcoordinate and a ycoordinate. The xcoordinate tells the number of units to the right of the origin (for positive numbers) or to the left of the origin (for negative numbers). The ycoordinate tells the number of units above the origin (for positive numbers) or below the origin (for negative numbers).
The coordinates of each point are often shown in what is called an ordered pairof numbers. An ordered pair looks like this: (2, 3). In every ordered pair, the first number is the xcoordinate, and the second number is the ycoordinate. So the ordered pair (2, 3) identifies a point with an xcoordinate of 2 and a ycoordinate of 3. The point is located at the intersection of the vertical line that is 2 units to the right of the origin (x = +2) and the horizontal line that is 3 units above the origin (y = +3). The point (2, –3) is located at the intersection of the vertical line that is 2 units to the right of the origin (x = +2) and the horizontal line that is 3 units below the origin (y = –3). The origin is identified by the ordered pair (0, 0).
The xand yaxes separate the graph into four parts called quadrants.
- Points in Quadrant I have positive numbers for both the xand the ycoordinates.
- Points in Quadrant II have a negative number for the xcoordinate but a positive number for the ycoordinate.
- Points in Quadrant III have negative numbers for both the xand the ycoordinates.
- Points in Quadrant IV have a positive number for the xcoordinate but a negative number for the ycoordinate.
Examples
The graph below shows the following points:
(2,3), (–3,2), (–4, –4), and (0,–2)
The graph below shows the following points:
A(4,–2), B(–1,1), C(3,3), and D(–4,–3).
Graphing Equations on the Coordinate Plane An equation with two variables xand ycan be graphed on a coordinate plane. Start by plugging in values for either xor y. Then solve the equation to find the value of the other variable. The xand y values make ordered pairs that you can plot on the graph.
Examples
Graph the equation x + y = 4
Solving for x, the equation becomes x = 4 – y
If y = 1, then x = 3.
If y = 2, then x = 2.
If y = 3, then x = 1.
If y = 4, then x = 0.
Plot the ordered pairs (3,1), (2,2), (1,3), and (0,4) on a graph. If you connect the points, you will see that the result is a straight line.
Graph the equation y – x2 = 2
y = 2 + x2
If x = 0, then y = 2.
If x = 1, then y = 3.
If x = 2, then y = 6.
If x = 3, then y = 11.
If x = 4, then y = 18.
Graph these points and connect the points with a line. Note that when you connect the points, you get a curved line.
Probability
When every event in a set of possible events has an equal chance of occurring, probability is the chance that a particular event (or "outcome") will occur. Probability is represented by the formula
Let's say you have a spinner with an arrow that spins around a circle that is divided into six equal parts. The parts are labeled from 1 to 6. When you spin the arrow, what is the probability that it will land on the part labeled 4? Following the formula:
Let's take that same spinner. What is the probability that the arrow will land on the number 2 or the number 3 when it is spun? Using the formula:
Examples
Solve: The National Fruit Growers' Association is conducting a random survey asking people to tell their favorite fruit. The chart shows the results so far.
What is the probability that the next randomly selected person will say that pears are his or her favorite fruit?
To solve probability problems, follow the word problem solution procedure outlined in Chapter 9.
Procedure
What must you find?The probability that a certain event will occur
What are the units?Fractions, decimals, or percents
What do you know?The number of people selecting each fruit as their favorite.
Create an equation and solve
Substitute values and solve.
Number of positive outcomes = 125 people who named pears as their favorite fruit
Number of possible outcomes = all people surveyed = 236 + 389 + 250 + 125 = 1,000
Solve: A box is filled with 25 black balls, 50 white balls, and 75 red balls. If Wendell reaches into the box and picks a ball without looking, what is the probability that he will pick a black or a white ball?
Procedure
What must you find?The probability that either of two events will occur
What are the units?Fractions, decimals, or percents
What do you know?How many of each kind of ball are in the box
Create an equation and solve.
Substitute values and solve.
Number of positive outcomes = number of black balls + number of white balls = 25 + 50 = 75
Number of possible outcomes = 25 + 50 + 75 = 150
Practice questions for this study guide can be found at:
Algebra and Probability Practice Problems for McGraw-Hill's ASVAB
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