Define and Evaluate Polynomials

Learning Outcomes

Define, evaluate, and simplify single variable polynomials

In the example on the previous page, we saw how combining the formulas for different shapes provides a way to accurately predict the amount of paint needed for a construction project. The result was a polynomial.

A polynomial function is a function consisting of the sum or difference of terms in which each term is a real number, a variable, or the product of a real number and variable(s) with a non-negative integer exponents. Non negative integers are $0, 1, 2, 3, 4$ , …
You may see a resemblance between expressions and polynomials which we have been studying in this course. 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	Reason
$2x^2-\frac{1}{2}x -9$	$\frac{2}{x^{2}}+x$	Polynomials only have variables in the numerator
$\frac{y}{4}-y^3$	$\frac{2}{y}+4$	Polynomials only have variables in the numerator
$\sqrt{12}\left(a\right)+9$	$\sqrt{a}+7$	Roots are equivalent to rational exponents, and polynomials only have integer exponents on variables

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 variable(s) with an exponent. The number part of the term is called the coefficient.

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

A polynomial containing two terms, such as $2x - 9$ , is called a binomial. A polynomial containing three terms, such as $-3{x}^{2}+8x - 7$ , is called a trinomial.

We can find the degree of a polynomial by identifying the highest power of the variable that occurs in the polynomial. The term with the highest degree is called the leading term because it is usually written first. The coefficient of the leading term is called the leading coefficient. When a polynomial is written so that the powers on a specified variable are descending, we say that it is in standard form. It is important to note that polynomials only have integer exponents.

4x^3 - 9x^2 + 6x, with the text "degree = 3" and an arrow pointing at the exponent on x^3, and the text "leading term =4" with an arrow pointing at the 4.

How to: Given a polynomial expression, identify the degree and leading coefficient

Find the highest power of x to determine the degree.
Identify the term containing the highest power of x to find the leading term.
Identify the coefficient of the leading term.

Example

For the following polynomials, identify the degree, the leading term, and the leading coefficient.

$3+2{x}^{2}-4{x}^{3}$
$5{t}^{5}-2{t}^{3}+7t$
$6p-{p}^{3}-2$

Show Solution

The highest power of x is $3$ , so the degree is $3$ . The leading term is the term containing that degree, $-4{x}^{3}$ . The leading coefficient is the coefficient of that term, $-4$ .
The highest power of t is $5$ , so the degree is $5$ . The leading term is the term containing that degree, $5{t}^{5}$ . The leading coefficient is the coefficient of that term, $5$ .
The highest power of p is $3$ , so the degree is $3$ . The leading term is the term containing that degree, $-{p}^{3}$ , The leading coefficient is the coefficient of that term, $-1$ .

In the following video, we will identify the terms, leading coefficient, and degree of a 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 $0$ does not 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 $0$ .

Evaluate a Polynomial

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 $-1$ 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 $-1$ .

$3+2+1$

Find the sum: $3+2+1=6$

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 $-3$ .

$-54 + 54 – 3=-3$

In the following video, we show more examples of evaluating polynomials for given values of the variable.

Module 8: Polynomials and Polynomial Functions

Learning Outcomes

How to: Given a polynomial expression, identify the degree and leading coefficient

Example

Evaluate a Polynomial

Example

Example

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