Polynomial Functions

You already know two kinds of polynomial functions very well: lines and parabolas. This page widens the view to cubics, quartics, and beyond. You’ll learn the vocabulary (degree, leading coefficient, constant term) and how to tell a polynomial apart from the exponential and sinusoidal functions you met in Grade 11.

Key ideas

Polynomial expressions

A polynomial expression is a sum of terms, where each term is a constant times a power of $x$ with a whole-number exponent ($0, 1, 2, 3, \dots$). For example:

$$x^3 - 5x^2 + 2x - 1$$

The constants in front ($1$, $-5$, $2$, $-1$) are the coefficients. They can be any real numbers, including fractions and decimals, or even numbers like $\sqrt{2}$. What matters is the exponents on $x$.

Polynomial functions

A polynomial function has the form

$$f(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0, \qquad a_n \ne 0$$

where $n$ is a whole number. Writing the terms from the highest power down is called standard form.

Word	Meaning	In $f(x) = -2x^4 + 3x^2 - x + 6$
degree	the highest exponent, $n$	$4$
leading coefficient	the coefficient of the highest power, $a_n$	$-2$
constant term	the term with no $x$, $a_0$ (also the $y$-intercept)	$6$

A polynomial is a function because each input $x$ gives exactly one output: you just multiply, add, and subtract. Its graph passes the vertical line test.

Names by degree

Degree	Name	Example
$0$	constant	$f(x) = 5$
$1$	linear	$f(x) = 2x - 3$
$2$	quadratic	$f(x) = x^2 - 4x + 1$
$3$	cubic	$f(x) = x^3 - 4x$
$4$	quartic	$f(x) = -x^4 + 2x^2$
$5$	quintic	$f(x) = x^5 - x$

So the linear and quadratic functions from earlier grades are simply polynomial functions of degree $1$ and $2$.

What is not a polynomial

An exponent that isn’t a whole number, or an $x$ somewhere other than a base, means it’s not a polynomial:

• $y = \sqrt{x} = x^{1/2}$: fractional exponent.
• $y = \dfrac{1}{x} = x^{-1}$: negative exponent.
• $y = 2^x$: the variable is in the exponent (this is exponential).
• $y = \sin x$: a trigonometric function (sinusoidal).

Comparing the graphs

A polynomial, an exponential, and a sinusoidal function have very different shapes.

Feature	Polynomial (degree 1 or more)	Exponential $y = 2^x$	Sinusoidal $y = \sin x$
domain	all real numbers	all real numbers	all real numbers
ends of the graph	both ends go to $+\infty$ or $-\infty$	one end flattens toward an asymptote	keeps oscillating, never settles
asymptotes	none	horizontal, $y = 0$	none
repeats?	no	no	yes, it’s periodic
$x$-intercepts	a limited number (at most the degree)	none	infinitely many

Polynomial graphs are smooth, unbroken curves with no asymptotes and no repeating pattern.

Worked examples

Example 1: Polynomial or not?

Decide whether each function is a polynomial function. If it is, state its degree and leading coefficient.

• (a) $f(x) = 4x^3 - 2x + 7$
• (b) $g(x) = \sqrt{x} + 1$
• (c) $h(x) = 5 - 3x^2 + \dfrac{1}{2}x^4$
• (d) $k(x) = 3^x$
• (e) $m(x) = \dfrac{2}{x^2}$

Solution.

(a) Yes. Every exponent is a whole number. Degree $3$, leading coefficient $4$.

(b) No. $\sqrt{x} = x^{1/2}$ has a fractional exponent.

(c) Yes. In standard form, $h(x) = \dfrac{1}{2}x^4 - 3x^2 + 5$. Degree $4$, leading coefficient $\dfrac{1}{2}$. A fraction as a coefficient is fine.

(d) No. The variable is in the exponent, so this is an exponential function.

(e) No. $\dfrac{2}{x^2} = 2x^{-2}$ has a negative exponent.

Example 2: Degree from factored form

State the degree, leading coefficient, and constant term of $f(x) = -2(x - 1)^2(x + 3)$ without expanding fully.

Solution. Multiply the highest power of $x$ from each factor: $(x)^2$ from $(x - 1)^2$ and $x$ from $(x + 3)$.

$$-2 \cdot x^2 \cdot x = -2x^3$$

So the degree is $3$ and the leading coefficient is $-2$.

The constant term is the value at $x = 0$:

$$f(0) = -2(0 - 1)^2(0 + 3) = -2(1)(3) = -6$$

Example 3: Writing in standard form

Write $f(x) = (2x - 1)^2 - x(x + 3)(x - 3)$ in standard form. Then state the degree, leading coefficient, and constant term.

Solution. Expand each part (see polynomial operations for a refresher):

$$\begin{aligned} (2x - 1)^2 &= 4x^2 - 4x + 1 \\ x(x + 3)(x - 3) &= x(x^2 - 9) = x^3 - 9x \end{aligned}$$

Subtract, being careful with the signs:

$$\begin{aligned} f(x) &= 4x^2 - 4x + 1 - x^3 + 9x \\ &= -x^3 + 4x^2 + 5x + 1 \end{aligned}$$

Degree $3$, leading coefficient $-1$, constant term $1$.

Check at $x = 1$: the original gives $(1)^2 - 1(4)(-2) = 1 + 8 = 9$, and the standard form gives $-1 + 4 + 5 + 1 = 9$. ✓

Example 4: Comparing key features

Compare $f(x) = x^3 - 4x$, $g(x) = 2^x$, and $h(x) = \sin x$ (with $x$ in degrees): give the range, the $x$-intercepts, and what happens at the ends of each graph.

Solution. Use the figure above.

• $f(x) = x^3 - 4x = x(x - 2)(x + 2)$. Its $x$-intercepts are $-2$, $0$, and $2$. As $x$ gets very large, $f(x)$ gets very large; as $x$ gets very negative, so does $f(x)$. Range $\{y \in \mathbb{R}\}$.
• $g(x) = 2^x$ has no $x$-intercepts, because $2^x \gt 0$ always. On the right it grows quickly; on the left it flattens toward the asymptote $y = 0$. Range $\{y \in \mathbb{R} \mid y \gt 0\}$.
• $h(x) = \sin x$ has infinitely many $x$-intercepts ($0^\circ$, $\pm 180^\circ$, $\pm 360^\circ$, …). It repeats every $360^\circ$ and never heads off to infinity. Range $\{y \in \mathbb{R} \mid -1 \le y \le 1\}$.

Only the polynomial heads to $\pm\infty$ at both ends, and it has only a few $x$-intercepts.

Common mistakes

Calling any function with powers of $x$ a polynomial. The exponents must be whole numbers. $y = x^{-1}$ and $y = x^{1/2}$ have powers of $x$, but they are not polynomials.

Confusing $x^2$ with $2^x$. In $x^2$ the variable is the base, so it’s a polynomial. In $2^x$ the variable is the exponent, so it’s exponential. Their graphs look nothing alike.

Reading the degree from the first term instead of the highest power. In $f(x) = 7 - 2x + 5x^4$, the degree is $4$ and the leading coefficient is $5$, even though $7$ is written first. Rewrite in standard form if it helps.

Forgetting to multiply all the factors when finding the degree. In $f(x) = (x - 1)^2(x + 3)$, the degree is $2 + 1 = 3$, not $2$. Add up the powers of $x$ from every factor.

Dropping a sign when subtracting a product. In Example 3, $-x(x^2 - 9)$ is $-x^3 + 9x$. Put the product in brackets before you subtract it.

Practice

1. (Warm-up) For $f(x) = 7 - 2x + 5x^4 - x^3$, state the degree, the leading coefficient, the constant term, and the name of the function by degree.

Solution

In standard form, $f(x) = 5x^4 - x^3 - 2x + 7$.

Degree $4$ (a quartic function), leading coefficient $5$, constant term $7$.

2. (Warm-up) Is each function a polynomial function? Explain.

• (a) $y = 3x^5 - x$
• (b) $y = x^{-2} + 1$
• (c) $y = \sqrt{2}\,x^2$
• (d) $y = 4^x$

Solution

(a) Yes: whole-number exponents $5$ and $1$.

(b) No: the exponent $-2$ is negative.

(c) Yes: the exponent is $2$. The coefficient $\sqrt{2}$ is just a real number, which is allowed.

(d) No: the variable is in the exponent, so it’s exponential.

3. (Warm-up) Name each function by its degree: (a) $y = 6 - x$, (b) $y = x^2(x - 1)$, (c) $y = 8$.

Solution

(a) Degree $1$: linear.

(b) $x^2 \cdot x = x^3$, so degree $3$: cubic.

(c) Degree $0$: constant.

4. (Core) Write $f(x) = (x + 2)(x - 1)(3x + 1)$ in standard form, and state its degree, leading coefficient, and constant term.

Solution

$$\begin{aligned} (x + 2)(x - 1) &= x^2 + x - 2 \\ (x^2 + x - 2)(3x + 1) &= 3x^3 + x^2 + 3x^2 + x - 6x - 2 \\ &= 3x^3 + 4x^2 - 5x - 2 \end{aligned}$$

Degree $3$, leading coefficient $3$, constant term $-2$.

Check: $f(0) = (2)(-1)(1) = -2$. ✓

5. (Core) Without expanding, find the degree, the leading coefficient, and the $y$-intercept of $g(x) = -(2x - 1)^3(x + 4)^2$.

Solution

The highest-power terms are $(2x)^3 = 8x^3$ and $x^2$:

$$-1 \cdot 8x^3 \cdot x^2 = -8x^5$$

Degree $5$, leading coefficient $-8$.

$$g(0) = -(-1)^3(4)^2 = -(-1)(16) = 16$$

The $y$-intercept is $16$.

6. (Core) Compare $y = x^2$ and $y = 2^x$. For each, give the range, the $y$-intercept, and what happens to $y$ as $x$ becomes very negative.

Solution

$y = x^2$: range $\{y \in \mathbb{R} \mid y \ge 0\}$, $y$-intercept $0$. As $x$ becomes very negative, $x^2$ becomes very large (for example, $(-100)^2 = 10\,000$).

$y = 2^x$: range $\{y \in \mathbb{R} \mid y \gt 0\}$, $y$-intercept $1$. As $x$ becomes very negative, $2^x$ gets closer and closer to $0$ (for example, $2^{-10} \approx 0.001$), so the graph flattens toward the asymptote $y = 0$.

7. (Core) Jaya says ”$y = \sin x$ can’t be a polynomial, because its graph crosses the $x$-axis too many times.” Explain what she means.

Solution

A polynomial of degree $n$ (with $n \ge 1$) has at most $n$ $x$-intercepts, so it can only cross the $x$-axis a limited number of times. The graph of $y = \sin x$ crosses the $x$-axis at every multiple of $180^\circ$, which is infinitely many times. No polynomial does that, so $\sin x$ isn’t a polynomial. (Also, its graph repeats and stays between $-1$ and $1$, while a non-constant polynomial heads to $\pm\infty$ at its ends.)

8. (Challenge) For what value of $a$ is $f(x) = (a - 2)x^3 + 4x^2 - 1$ a quadratic function? What is its leading coefficient then?

Solution

The $x^3$ term must disappear, so $a - 2 = 0$ and $a = 2$.

Then $f(x) = 4x^2 - 1$, a quadratic with leading coefficient $4$.

9. (Challenge) The relation $x = y^2 - 4$ uses only whole-number powers. Is $y$ a polynomial function of $x$? Explain.

Solution

No. It isn’t even a function of $x$. For example, at $x = 0$ we get $y^2 = 4$, so $y = 2$ or $y = -2$: one input gives two outputs. (Its graph is a sideways parabola, which fails the vertical line test.) A polynomial function must have the form $y =$ (a polynomial in $x$).