# Polynomial Functions: Degree, Graphs, Derivatives and Integrals

> Polynomial functions explained: degree, leading term, end behavior, roots and turning points, plus how to differentiate and integrate any polynomial.

Source: https://practicecalculus.com/functions/polynomial-functions
Updated: 2026-11-03

$$ f(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0 $$

## At a glance

- **Domain:** $(-\infty, \infty)$
- **Derivative:** $na_nx^{n-1} + (n-1)a_{n-1}x^{n-2} + \cdots + a_1$
- **Integral:** $\frac{a_n}{n+1}x^{n+1} + \cdots + \frac{a_1}{2}x^2 + a_0x + C$
- **Continuity:** Continuous and differentiable everywhere
- **Real roots:** At most $n$
- **Turning points:** At most $n - 1$

## Key takeaways

- A polynomial is a sum of terms $a_kx^k$ with whole-number exponents; its degree is the highest exponent with a nonzero coefficient.
- Differentiate and integrate a polynomial one term at a time with the power rule.
- Each derivative lowers the degree by one, and each integral raises it by one.
- A degree $n$ polynomial has at most $n$ real roots and at most $n - 1$ turning points.

A **polynomial function** is a finite sum of power functions with whole-number exponents, each multiplied by a constant. Lines, parabolas and cubics are all polynomials. They are the friendliest functions in calculus: continuous everywhere, smooth everywhere, and differentiated or integrated term by term.

## Definition and formula

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

- The **degree** is $n$, the highest power with a nonzero coefficient.
- The **leading term** is $a_nx^n$ and the **leading coefficient** is $a_n$.
- The **constant term** $a_0$ is the $y$-intercept.

Exponents must be whole numbers $0, 1, 2, \ldots$. So $x^3 - 2x$ is a polynomial, but $\sqrt{x}$, $\frac{1}{x}$ and $2^x$ are not.

| Degree | Name | Example |
| --- | --- | --- |
| 0 | [constant](/functions/constant-function) | $5$ |
| 1 | [linear](/functions/linear-function) | $2x - 1$ |
| 2 | [quadratic](/functions/quadratic-function) | $x^2 + 3x$ |
| 3 | cubic | $x^3 - 3x$ |
| 4 | quartic | $x^4 - x^2$ |

## Graph of a polynomial

```graph
title: 'Graph of the cubic polynomial y = x³ - 3x and its derivative'
caption: 'The cubic has turning points at $x = -1$ and $x = 1$, where its derivative $3x^2 - 3$ is zero.'
x: [-2.5, 2.5]
y: [-4, 5]
functions:
  - expr: x^3 - 3x
    label: y = x³ - 3x
  - expr: 3x^2 - 3
    label: y = 3x² - 3 (derivative)
    dashed: true
points:
  - [-1, 2, 'max (-1, 2)']
  - [1, -2, 'min (1, -2)']
```

## Key properties

- **Continuous and differentiable everywhere.** No holes, jumps, corners or asymptotes, so every limit is found by [direct substitution](/rules/direct-substitution).
- **End behavior** comes from the leading term alone. Even degree with $a_n > 0$: both ends up. Odd degree with $a_n > 0$: down on the left, up on the right. A negative $a_n$ flips the picture.
- **Roots.** A degree $n$ polynomial has at most $n$ real roots. An odd-degree polynomial always has at least one, by the [intermediate value theorem](/theorems/intermediate-value-theorem).
- **Turning points.** At most $n - 1$, because $f'$ has degree $n - 1$.

## Derivative of a polynomial

Combine the [power rule](/rules/power-rule), [constant multiple rule](/rules/constant-multiple-rule) and [sum rule](/rules/sum-and-difference-rule):

$$f'(x) = na_nx^{n-1} + (n-1)a_{n-1}x^{n-2} + \cdots + 2a_2x + a_1$$

The constant term disappears and the degree drops by one. Differentiating $n + 1$ times gives $0$.

## Integral of a polynomial

$$\int f(x)\,dx = \frac{a_n}{n+1}x^{n+1} + \cdots + \frac{a_1}{2}x^2 + a_0x + C$$

The degree rises by one. Definite integrals of polynomials are always easy to evaluate with the [fundamental theorem of calculus](/blog/fundamental-theorem-of-calculus).

## Worked examples

**Example 1: derivative.** $\frac{d}{dx}\left(4x^5 - 2x^3 + 7x - 1\right) = 20x^4 - 6x^2 + 7$.

**Example 2: turning points.** For $f(x) = x^3 - 3x$, $f'(x) = 3x^2 - 3 = 3(x-1)(x+1)$, which is zero at $x = \pm 1$. Then $f(-1) = 2$ is a local maximum and $f(1) = -2$ is a local minimum.

**Example 3: integral.** $\int (6x^2 - 4x + 3)\,dx = 2x^3 - 2x^2 + 3x + C$.

**Example 4: definite integral.** $\int_0^2 (x^3 - x)\,dx = \left[\frac{x^4}{4} - \frac{x^2}{2}\right]_0^2 = (4 - 2) - 0 = 2$.

## Common mistakes

1. **Differentiating a product without expanding or using the product rule.** $\frac{d}{dx}\left[(x^2 + 1)(x - 3)\right]$ is not $2x \cdot 1$. Expand first or use the [product rule](/rules/product-rule).
2. **Keeping the constant term in a derivative.** $\frac{d}{dx}(x^2 + 5) = 2x$, not $2x + 5$.
3. **Forgetting $+\,C$** on indefinite integrals.
4. **Judging end behavior from the first term written.** Use the term of highest degree, wherever it appears.

## How it connects

Dividing two polynomials gives the [rational functions](/functions/rational-functions). Taylor polynomials approximate other functions by polynomials, and [Taylor series](/blog/taylor-series) push this to infinitely many terms. For the rules behind each step, see [power functions](/functions/power-functions).

## Frequently asked questions

### How do you find the derivative of a polynomial?

Apply the power rule to each term and add the results. For example, $\frac{d}{dx}(2x^3 - 5x^2 + x - 4) = 6x^2 - 10x + 1$.

### How do you integrate a polynomial?

Use the reverse power rule on each term: add one to the exponent and divide by the new exponent, then add $C$. For example, $\int (3x^2 + 4x)\,dx = x^3 + 2x^2 + C$.

### What determines the end behavior of a polynomial?

Only the leading term $a_nx^n$ matters as $x \to \pm\infty$. Even degree means both ends go the same way; odd degree means they go opposite ways, and the sign of $a_n$ decides which.

### Are polynomials always continuous?

Yes. Every polynomial is continuous and differentiable at every real number, so limits of polynomials can always be found by direct substitution.

### How many turning points can a polynomial have?

A polynomial of degree $n$ has at most $n - 1$ turning points, because its derivative has degree $n - 1$ and therefore at most $n - 1$ zeros.