Algebra and Probability Study Guide 2 for McGraw-Hill's ASVAB (page 3)

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:

x² + 5x + 6

First, you know that x times x will give x², so it is likely that each factor is going to start with x.

(x   ) (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

(x 2) (x 3)

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

(x + 2)(x + 3)

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 + 8x = 8

6x² can be factored into either (6x)(x) or (2x)(3x). Using the latter, the first terms in our factors are as follows:

(2x ) (3x )

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:

(2x   2)(3x   4)

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: