Define and Evaluate Polynomials

Learning Objectives

Identify the terms, the coefficients, and the exponents of a polynomial
Evaluate a polynomial for given values of the variable

Identify the terms, the coefficients, and the exponents of a polynomial

Polynomials are algebraic expressions that are created by combining numbers and variables using arithmetic operations such as addition, subtraction, multiplication, division, and exponentiation. You can create a polynomial by adding or subtracting terms. Polynomials are very useful in applications from science and engineering to business. You may see a resemblance between expressions, which we have been studying in this course, and polynomials. Polynomials are a special sub-group of mathematical expressions and equations.

The following table is intended to help you tell the difference between what is a polynomial and what is not.

IS a Polynomial	Is NOT a Polynomial	Because
$2x^2-\frac{1}{2}x -9$	$\frac{2}{x^{2}}+x$	Polynomials only have variables in the numerator
$\frac{y}{4}-y^3$	$\frac{2}{y}+4$	Polynomials only have variables in the numerator
$\sqrt{12}\left(a\right)+9$	$\sqrt{a}+7$	Variables under a root are not allowed in polynomials

The basic building block of a polynomial is a monomial. A monomial is one term and can be a number, a variable, or the product of a number and variables with an exponent. The number part of the term is called the coefficient.

Examples of monomials:

number: ${2}$
variable: ${x}$
product of number and variable: ${2x}$
product of number and variable with an exponent: ${2x}^{3}$

The expression 6x to the power of 3. 6 is the coefficient, x is the variable, and the power of 3 is the exponent.

The coefficient can be any real number, including 0. The exponent of the variable must be a whole number—0, 1, 2, 3, and so on. A monomial cannot have a variable in the denominator or a negative exponent.

The value of the exponent is the degree of the monomial. Remember that a variable that appears to have no exponent really has an exponent of 1. And a monomial with no variable has a degree of 0. (Since $x^{0}$ has the value of 1 if $x\neq0$ , a number such as 3 could also be written $3x^{0}$ , if $x\neq0$ as $3x^{0}=3\cdot1=3$ .)

Example

Identify the coefficient, variable, and degree of the variable for the following monomial terms:
1) 9
2) x
3) $\displaystyle \frac{3}{5}{{k}^{8}}$

Show Solution

1)
9 is a constant so there is no coefficient or variable. Since there is no variable, we consider the degree to be 0.

2)
The variable is x.

The exponent of x is 1. $x=x^{1}$ , so the degree is 1.

The coefficient of x is 1. $x=1x^{1}$ .

3)
The variable is k.

The exponent of k is 8, so the degree is 8.

The coefficient of $k^{8}$ is $\displaystyle \frac{3}{5}$

A polynomial is a monomial or the sum or difference of two or more polynomials. Each monomial is called a term of the polynomial.

Some polynomials have specific names indicated by their prefix.

monomial—is a polynomial with exactly one term (“mono”—means one)
binomial—is a polynomial with exactly two terms (“bi”—means two)
trinomial—is a polynomial with exactly three terms (“tri”—means three)

The word “polynomial” has the prefix, “poly,” which means many. However, the word polynomial can be used for all numbers of terms, including only one term.

Because the exponent of the variable must be a whole number, monomials and polynomials cannot have a variable in the denominator.

Polynomials can be classified by the degree of the polynomial. The degree of a polynomial is the degree of its highest degree term. So the degree of $2x^{3}+3x^{2}+8x+5$ is 3.

A polynomial is said to be written in standard form when the terms are arranged from the highest degree to the lowest degree. When it is written in standard form it is easy to determine the degree of the polynomial.

The table below illustrates some examples of monomials, binomials, trinomials, and other polynomials. They are all written in standard form.

Monomials	Binomials	Trinomials	Other Polynomials
15	$3y+13$	$x^{3}-x^{2}+1$	$5x^{4}+3x^{3}-6x^{2}+2x$
$\displaystyle \frac{1}{2}x$	$4p-7$	$3x^{2}+2x-9$	$\frac{1}{3}x^{5}-2x^{4}+\frac{2}{9}x^{3}-x^{2}+4x-\frac{5}{6}$
$-4y^{3}$	$3x^{2}+\frac{5}{8}x$	$3y^{3}+y^{2}-2$	$3t^{3}-3t^{2}-3t-3$
$16n^{4}$	$14y^{3}+3y$	$a^{7}+2a^{5}-3a^{3}$	$q^{7}+2q^{5}-3q^{3}+q$

When the coefficient of a polynomial term is 0, you usually do not write the term at all (because 0 times anything is 0, and adding 0 doesn’t change the value). The last binomial above could be written as a trinomial, $14y^{3}+0y^{2}+3y$ .

A term without a variable is called a constant term, and the degree of that term is 0. For example 13 is the constant term in $3y+13$ . You would usually say that $14y^{3}+3y$ has no constant term or that the constant term is 0.

Example

For the following expressions, determine whether they are a polynomial. If so, categorize them as a monomial, binomial, or trinomial.

$\frac{x-3}{1-x}+x^2$
$t^2+2t-3$
$x^3+\frac{x}{8}$
$\frac{\sqrt{y}}{2}-y-1$

Show Solution

$\frac{x-3}{1-x}+x^2$ is not a polynomial because it violates the rule that polynomials cannot have variables in the denominator of a fraction.
$t^2+2t-3$ is a polynomial because it is an expression whose terms are connected by addition and subtraction, and there are no variables under a root or in the denominator of a fraction. There are three terms in this polynomial so it is a trinomial.
$x^3+\frac{x}{8}$ is a polynomial because it is an expression whose terms are connected by addition and subtraction, and there are no variables under a root or in the denominator of a fraction. There are two terms in this polynomial so it is a binomial.
$\frac{\sqrt{y}}{2}-y-1$ is not a polynomial because it violates the rule that polynomials cannot have variables under a root.

In the following video, you will be shown more examples of how to identify and categorize polynomials.

Evaluate a polynomial for given values of the variable

You can evaluate polynomials just as you have been evaluating expressions all along. To evaluate an expression for a value of the variable, you substitute the value for the variable every time it appears. Then use the order of operations to find the resulting value for the expression.

Example

Evaluate $3x^{2}-2x+1$ for $x=-1$ .

Show Solution

Substitute

- 1

$-1$ for each x in the polynomial.

$3\left(-1\right)^{2}-2\left(-1\right)+1$

Following the order of operations, evaluate exponents first.

$3\left(1\right)-2\left(-1\right)+1$

Multiply 3 times 1, and then multiply $-2$ times $-1$ .

$3+\left(-2\right)\left(-1\right)+1$

Change the subtraction to addition of the opposite.

$3+2+1$

Find the sum.

Answer

$3x^{2}-2x+1=6$ , for $x=-1$

Example

Evaluate $\displaystyle -\frac{2}{3}p^{4}+2p^{3}-p$ for $p = 3$ .

Show Solution

Substitute 3 for each p in the polynomial.

$\displaystyle -\frac{2}{3}\left(3\right)^{4}+2\left(3\right)^{3}-3$

Following the order of operations, evaluate exponents first and then multiply.

$\displaystyle -\frac{2}{3}\left(81\right)+2\left(27\right)-3$

Add and then subtract to get $-3$ .

$-54 + 54 – 3$

Answer

$\displaystyle -\frac{2}{3}p^{4}+2p^{3}-p=-3$ , for $p = 3$

The following video presents more examples of evaluating a polynomial for a given value.

Objective 3

Learning Objectives

Identify the terms, the coefficients, and the exponents of a polynomial

Example

Example

Evaluate a polynomial for given values of the variable

Example

Answer

Example

Answer

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