Polynomial Function

Mapping Inputs to Polynomial Values

Each input value produces exactly one output value. With the polynomial expression $2x^2 - 3x + 1$ as its rule, the function is written $P(x) = 2x^2 - 3x + 1$ . Substituting the input $x = 2$ and evaluating the powers first returns $P(2) = 3$ .

General Form of a Polynomial Function

A polynomial function in the variable $x$ is generally written in the form:

\begin{aligned} P(x) &= a_n x^n + a_{n-1} x^{n-1} \\ &\quad{}+ a_{n-2} x^{n-2} + \dots + a_1 x + a_0 \end{aligned}

The general form has the following components:

$P(x)$ :

Function notation, read "P of x", indicating the function's value depends on the value of $x$ .
$x$ :

The variable of the polynomial function.
$n$ :

The highest power of the variable $x$ . This value $n$ must be a non-negative integer ( $0, 1, 2, 3, \dots$ ). This non-negative integer $n$ also determines the degree of the polynomial function.
$a_n, a_{n-1}, \dots, a_1, a_0$ :

The coefficients of the polynomial function. A real polynomial function has real-number coefficients.
$a_n x^n$ :

The term with the highest power. This term is called the leading term.
$a_n$ :

The leading coefficient $a_n$ must be nonzero, so $a_n\neq0$ , for the function to have degree $n$ .
$a_0$ :

The term without the variable $x$ (or can be considered $a_0 x^0$ ). This term is called the constant term or constant.

Example of a Polynomial Function

Suppose we have the function: $f(x) = 5x^3 - 2x^2 + 7x - 1$

This is a polynomial function in the variable $x$ .
Its degree is $3$ (the highest power of $x$ ).
Its leading term is $5x^3$ .
Its leading coefficient is $5$ ( $a_3 = 5$ ).
Other coefficients are $a_2 = -2$ , $a_1 = 7$ .
Its constant term is $-1$ ( $a_0 = -1$ ).

A polynomial function has nonnegative integer exponents and a nonzero leading coefficient.

Exercise

p(t)=-2t^4+3t^2-t+8

Identify its degree, leading term, leading coefficient, coefficient of $t^3$ , and constant term. Then calculate $p(2)$ .

Worked Solution

Write the missing power explicitly:

p(t)=-2t^4+0t^3+3t^2-t+8

Degree: $4$
Leading term: $-2t^4$
Leading coefficient: $-2$
Coefficient of $t^3$ : $0$
Constant term: $8$

Evaluate the function by substituting $t=2$ :

p(2)=-2(2)^4+3(2)^2-2+8

Mapping Inputs to Polynomial Values

Each input value produces exactly one output value. With the polynomial expression

2x^2 - 3x + 1

as its rule, the function is written

P(x) = 2x^2 - 3x + 1

. Substituting the input

x = 2

and evaluating the powers first returns

P(2) = 3

General Form of a Polynomial Function

A polynomial function in the variable

x

is generally written in the form:

\begin{aligned} P(x) &= a_n x^n + a_{n-1} x^{n-1} \\ &\quad{}+ a_{n-2} x^{n-2} + \dots + a_1 x + a_0 \end{aligned}

The general form has the following components:

$P(x)$ :

Function notation, read "P of x", indicating the function's value depends on the value of $x$ .

$x$ :

The variable of the polynomial function.

$n$ :

The highest power of the variable $x$ . This value $n$ must be a non-negative integer ( $0, 1, 2, 3, \dots$ ). This non-negative integer $n$ also determines the degree of the polynomial function.

$a_n, a_{n-1}, \dots, a_1, a_0$ :

The coefficients of the polynomial function. A real polynomial function has real-number coefficients.

$a_n x^n$ :

The term with the highest power. This term is called the leading term.

$a_n$ :

The leading coefficient $a_n$ must be nonzero, so $a_n\neq0$ , for the function to have degree $n$ .

$a_0$ :

The term without the variable $x$ (or can be considered $a_0 x^0$ ). This term is called the constant term or constant.

Example of a Polynomial Function

Suppose we have the function:

f(x) = 5x^3 - 2x^2 + 7x - 1

This is a polynomial function in the variable

x

Its degree is

3

(the highest power of

x

Its leading term is

5x^3

Its leading coefficient is

5

(

a_3 = 5

Other coefficients are

a_2 = -2

a_1 = 7

Its constant term is

-1

(

a_0 = -1

A polynomial function has nonnegative integer exponents and a nonzero leading coefficient.

Exercise

p(t)=-2t^4+3t^2-t+8

Identify its degree, leading term, leading coefficient, coefficient of

t^3

, and constant term. Then calculate

p(2)

Write the missing power explicitly:

p(t)=-2t^4+0t^3+3t^2-t+8

Degree:

4

Leading term:

-2t^4

Leading coefficient:

-2

Coefficient of

t^3

0

Constant term:

8

Evaluate the function by substituting

t=2

p(2)=-2(2)^4+3(2)^2-2+8

=-32+12-2+8

=-14