High School Algebra I Unlocked (2016)

GOALS

By the end of this chapter, you will be able to:

•Define and explain polynomial expressions

•Combine polynomial expressions through addition and subtraction

•Multiply polynomial expressions using the distributive property and FOIL

•Factor polynomial expressions by reversing the FOIL process or by completing the square

Lesson 3.1. Introduction to Polynomial Expressions

REVIEW

BEFORE BEGINNING THIS CHAPTER, YOU SHOULD BE FAMILIAR WITH:

•general number properties

•the parts of an equation or expression

•basic mathematical operations

•solving numeric and variable expressions

•the rules of exponents and manipulating exponential terms

In the first two chapters, we discussed the roles of both variables and exponents. Now we’ll discuss polynomial expressions, which result when the world of variables collides with the world of exponents.

A single-variable polynomial is an expression of the sum of two or more terms that contain different powers of the same variable. All of the following are single-variable polynomial expressions:

In Latin, the term poly
means “many” and
nomial means “term.”
Therefore, a polynomial
is “many terms.”

A polynomial can have 2 terms, 20 terms, or 200 terms. Polynomials having one, two, or three terms are referred to as monomials, binomials, and trinomials, respectively.

The degree of a single-variable polynomial is equal to the largest exponent associated with that variable. Take a look at the following table to see how the degree of a polynomial relates to the polynomial itself:

Polynomials can also be named for their degree: A quadratic is a second-degree polynomial, a cubic is a third-degree polynomial, and a quartic is a fourth-degree polynomial. In algebra, you will focus quite a bit on quadratics, particularly quadratic equations, which are often used to represent rates of change. But you don’t have to worry about this yet—we will tackle quadratic equations in more detail in Chapter 6.

A polynomial with two variables is an expression of the sum of more than two terms that contain different powers of the same variables. The following are polynomial expressions with two variables:

The degree of a term in a multi-variable polynomial is equal to the sum of the exponents in that term, and the degree of a multi-variable polynomial is the greatest sum. The degrees of the two-variable polynomial expressions are shown in the following table.

When working with
polynomials, you will
encounter variables
without coefficients, such
as x, y², and z³. If a variable
does not have a coefficient
in front of it, the coefficient
is assumed to be 1.

To express polynomials in standard form, you put the terms in order of degree and combine like terms. (See Lesson 3.2 for more on like terms.) You will notice that all of the previous polynomial expressions place the terms in descending order according to the term’s degree. For example, in y² − y + 6, the term y² has a degree of 2 and is placed first; the term y has a degree of 1 and is placed second; and the final term, 6, is a constant with no degree, and is placed last.

While you may not always be required to express polynomials in standard form, doing so helps organize the terms and, in turn, reduces errors stemming from adding, subtracting, multiplying, or dividing terms.

Finally, there are some characteristics of polynomials that you should be familiar with.

Now that we’ve had a crash course in polynomials, let’s see if we can answer a few questions about them.

EXAMPLE

Which of the following is a polynomial expression?

A) 2x⁴ + x⁻⁶ + 4

B) + 2x² + 3

C) x⁸ + 10x² + 1

D) 4x⁴ +

To answer this question, you need to understand the properties of polynomials, so take a look at each polynomial expression to see if it fits the criteria. Choice (A) gives you the expression 2x⁴ + x⁻⁶ + 4. While a polynomial can have terms with coefficients, variables, exponents, and constants, a polynomial cannot have a term with a negative exponent. Therefore, the term x⁻⁶ does not allow the expression to be classified as a polynomial; eliminate (A). The expression in (B), + 2x² + 3, has a term with variables in the denominator, ; since polynomials cannot have variables in the denominator, eliminate (B). The expression in (C), x⁸ + 10x² + 1, has terms that adhere to the rules of polynomials; keep (C) as an option. Finally, the expression in (D), 4x⁴ + , contains a term with a variable under a radical, , which prohibits the expression from being classified as a polynomial. Therefore, the only expression that is a polynomial expression is (C), x⁸ + 10x² + 1.

EXAMPLE

What is the degree of the polynomial 2x⁴ + x⁶ + 4 ?

In order to answer this question, you must know that the degree of a single-variable polynomial is equal to the largest exponent associated with that variable. In the expression 2x⁴ + x⁶ + 4, the term x⁶ is the term with the largest exponent. Therefore, the degree of the expression 2x⁴ + x⁶ + 4 is 6.

Now let’s look at a question that deals with expressing polynomials in standard form.

EXAMPLE

Express the polynomial 4 + 2x⁴ + x⁶ in standard form.

To express polynomials in standard form, you need to put the terms in order of degree from highest to lowest. In the expression 4 + 2x⁴ + x⁶, the term x⁶ has a degree of 6, the term 2x⁴ has a degree of 4, and the term 4 is a constant and has a degree of 0. Therefore, the polynomial 4 + 2x⁴ + x⁶ would be expressed in standard form as x⁶ + 2x⁴ + 4.

Let’s do one more question before moving on.

EXAMPLE

Evaluate 4 + 2x⁴ + x⁶ when x = 2.

Are you thinking to yourself, “Wait a minute! You never mentioned anything about evaluating polynomials!” Don’t panic when you see a question like this. You evaluate polynomials in the same way that you evaluate other expressions. Here, they ask you to evaluate 4 + 2x⁴ + x⁶ when x = 2, so replace x with 2 in the expression and solve:

4 + 2x⁴ + x⁶ = 4 + 2(2)⁴ + (2)⁶

   = 4 + 2(16) + 64

  = 4 + 32 + 64

= 100

So, when x = 2, the value of 4 + 2x⁴ + x⁶ is 100.

See? Not too bad, right?