---
title: How Many Real Roots Does a Cubic Have?
subtitle: Reading the answer off the derivative of ax³ + bx² + cx + d
author: Mathematics Department
organization: Your School
description: A short, rigorous handout showing why every real cubic has one, two, or three distinct real roots, and how the sign of the product of its two stationary values decides which.
keywords: [cubic polynomial, real roots, derivative, stationary points, first derivative test, end behavior, repeated root, discriminant]
template: assignment
paper: us-letter
titlePage: false
toc: false
headingNumbers: false
equationNumbers: true
columns: 1
figurePlacement: none
tableCaptionPosition: top
header: Real Roots of a Cubic | Derivative Method
footer: Handout - counting real roots from stationary values
pdfProfile: tagged
language: en-US
---

# 1. The ends belong to the leading term

Throughout, $f(x)=ax^{3}+bx^{2}+cx+d$ has real coefficients and $a\ne0$. *Root* always means real root, and *how many roots* always means how many **distinct** real numbers satisfy $f(x)=0$. For $x\ne0$ factor out the leading term:

:::equation{#eq:dominance alt="f of x equals a x cubed times the quantity one plus b over a times one over x plus c over a times one over x squared plus d over a times one over x cubed"}
\[
f(x)=ax^{3}\left(1+\frac{b}{a}\cdot\frac1x+\frac{c}{a}\cdot\frac1{x^{2}}+\frac{d}{a}\cdot\frac1{x^{3}}\right).
\]
:::

Every term in the bracket except the $1$ carries a power of $x$ in its denominator, so the bracket approaches $1$ as $\lvert x\rvert$ grows and $f$ behaves like $ax^{3}$ far from the origin: the lower-degree coefficients bend the curve near the origin but cannot alter what it does at the two ends. If $a>0$ the graph falls to $-\infty$ on the left and rises to $+\infty$ on the right; if $a<0$ the ends are exchanged.

> [!NOTE]
> **A concrete bound.** Put $M=\max\bigl(\lvert b\rvert,\lvert c\rvert,\lvert d\rvert\bigr)/\lvert a\rvert$. Whenever $\lvert x\rvert>1+M$ the remaining terms are together smaller than $M\bigl(\lvert x\rvert^{2}+\lvert x\rvert+1\bigr)<(\lvert x\rvert-1)\bigl(\lvert x\rvert^{2}+\lvert x\rvert+1\bigr)=\lvert x\rvert^{3}-1$. So $f(x)$ takes the sign of $ax^{3}$ there, and every real root of $f$ lies in $[-1-M,\,1+M]$.

Two consequences follow at once. Because $f$ is continuous and takes both signs, the Intermediate Value Theorem gives **at least one** real root; a cubic has at most three roots, so there are **at most three**. Whether the answer is one, two or three is settled between the ends, where the derivative rules.

# 2. Stationary points and the quantity $b^{2}-3ac$

Differentiating produces a quadratic:

:::equation{#eq:derivative alt="f prime of x equals three a x squared plus two b x plus c, with discriminant four times delta where delta is b squared minus three a c"}
\[
f'(x)=3ax^{2}+2bx+c,\qquad
4b^{2}-12ac=4\delta,\qquad
\delta:=b^{2}-3ac.
\]
:::

If $\delta<0$ then $f'$ has no real zero and keeps the sign of $a$ throughout, so $f$ is strictly monotone. If $\delta=0$ then $f'(x)=3a\bigl(x+\tfrac{b}{3a}\bigr)^{2}$ vanishes only at $x_{0}=-b/(3a)$, and expanding about $x_{0}$ gives $f(x)=a(x-x_{0})^{3}+f(x_{0})$, a translated copy of $ax^{3}$, again strictly monotone. A strictly monotone function attains every value exactly once, so:

> [!IMPORTANT]
> If $\delta\le0$ the cubic is strictly monotone and has **exactly one** real root, whatever $d$ may be.

If $\delta>0$ then $f'$ has two distinct zeros, the **stationary points**

:::equation{#eq:stationary alt="p equals minus b minus the square root of delta over three a, and q equals minus b plus the square root of delta over three a"}
\[
p=\frac{-b-\sqrt{\delta}}{3a},\qquad q=\frac{-b+\sqrt{\delta}}{3a}.
\]
:::

The derivative changes sign at $p$ and at $q$ and nowhere else, so $f$ is monotone on each of the three intervals they cut out, and one of $f(p)$, $f(q)$ is a local maximum and the other a local minimum. Which is which depends on the sign of $a$; @tbl:rule never needs to know.

# 3. The counting rule

Suppose $\delta>0$ and take $a>0$, so $p<q$ with $f(p)$ the local maximum. Reading left to right, $f$ climbs from $-\infty$ to $f(p)$, descends to $f(q)$, then climbs to $+\infty$; each branch is strictly monotone and meets the level $0$ at most once. Since $f$ strictly decreases between $p$ and $q$, $f(p)>f(q)$ and the two stationary values never vanish together. Reversing the sign of $a$ reflects the picture without changing any product, so the rule holds for either sign.

:::table{#tbl:rule caption="The number of distinct real roots of a cubic, decided by two quantities taken from the derivative"}
| Condition | Shape of the graph | Roots |
| --- | --- | ---: |
| $\delta\le0$ | strictly monotone; no turning point | 1 |
| $\delta>0$ and $f(p)\,f(q)>0$ | both turning values on the same side of the axis | 1 |
| $\delta>0$ and $f(p)\,f(q)=0$ | a turning point sits on the axis, giving a repeated root | 2 |
| $\delta>0$ and $f(p)\,f(q)<0$ | the axis passes between the turning values | 3 |
:::

:::figure{#fig:family caption="One shape, three positions of the axis: y = x³ − 3x + k for k = 0, 2 and 4, with stationary points at x = −1 and x = 1" alt="Three panels of the same cubic curve with the horizontal axis drawn at three different heights, meeting the curve three times, twice, and once" width=76 placement=none}
![Three panels showing one cubic curve meeting the horizontal axis three times, twice, and once](cubic-root-cases.svg)
:::

@fig:family displays the classification in a single family: the curve never changes, only the height of the axis, and the stationary values $f(-1)=k+2$ and $f(1)=k-2$ move with it.

> [!TIP]
> Repeated roots are exactly the roots shared by $f$ and $f'$: with $\delta>0$ that is the third line of @tbl:rule, and with $\delta=0$ it happens only when $f(x_{0})=0$, where $f(x)=a(x-x_{0})^{3}$ has a triple root — still one distinct value.

**Remark.** Substituting $x=t-b/(3a)$ removes the quadratic term, so after dividing by $a$ every cubic reduces to $t^{3}+pt+q$, with stationary points $t=\pm\sqrt{-p/3}$. Since $t^{3}+pt+q=t(t^{2}+p)+q$, the stationary values multiply to $q^{2}+\tfrac{4}{27}p^{3}$: the rule above is the classical discriminant $\Delta=-4p^{3}-27q^{2}$ in disguise, with $\Delta>0$ for three roots, $\Delta=0$ for a repeated root, and $\Delta<0$ for one.

# 4. Worked examples

**Example 1.** $f(x)=x^{3}-3x+1$ has $\delta=0-3(1)(-3)=9>0$, and $f'(x)=3x^{2}-3$ vanishes at $x=\pm1$. The stationary values $f(-1)=3$ and $f(1)=-1$ have product $-3$, so there are **three** distinct real roots, and the sign changes at $f(-2)=-1$, $f(0)=1$, $f(2)=3$ place them in $(-2,-1)$, $(0,1)$ and $(1,2)$.

**Example 2.** $f(x)=x^{3}-3x+5$ has the same stationary points, but $f(-1)=7$ and $f(1)=3$. The product $21$ is positive, so there is **one** real root, and since $f(-3)=-13$ it lies in $(-3,-1)$. Two turning points do not by themselves produce three roots.

**Example 3.** $f(x)=x^{3}-3x+2$ gives $f(-1)=4$ and $f(1)=0$. The product vanishes, so $x=1$ is a repeated root and there are **two** distinct real roots; division confirms $f(x)=(x-1)^{2}(x+2)$.

**Example 4.** $f(x)=x^{3}+x^{2}+x+1$ has $\delta=1^{2}-3(1)(1)=-2<0$, so $f$ is strictly increasing and has **one** real root. The constant term was never used; the factorisation $f(x)=(x+1)(x^{2}+1)$ confirms the root $x=-1$.

**Example 5.** $f(x)=-2x^{3}+3x^{2}+12x-4$ has $\delta=3^{2}-3(-2)(12)=81>0$ and $f'(x)=-6(x-2)(x+1)$, so the stationary points are $-1$ and $2$, with $f(-1)=-11$ and $f(2)=16$. The product $-176$ is negative, so there are **three** distinct real roots, and $a<0$ makes $x=-1$ the local minimum, which the test never needed to know.

# 5. Recovering a parameter from the root count

**Example 6.** For which real $k$ does $x^{3}-3x+k$ have three distinct real roots? The derivative does not involve $k$, so the stationary points stay at $\pm1$ and the stationary values are $k+2$ and $k-2$. Their product $k^{2}-4$ is negative exactly when $-2<k<2$, the condition for three roots; it vanishes at $k=\pm2$, giving two distinct roots, and is positive for $\lvert k\rvert>2$, giving one. The panels of @fig:family are $k=0$, $2$ and $4$.

**Example 7.** For which real $k$ does $x^{3}-6x^{2}+9x+k$ have a repeated root, and what are the roots? Here $\delta=(-6)^{2}-3(1)(9)=9>0$ and $f'(x)=3(x-1)(x-3)$, so the stationary values are $f(1)=k+4$ and $f(3)=k$, and a repeated root needs $k(k+4)=0$. Then $k=0$ gives $f(x)=x(x-3)^{2}$ with roots $0$ and $3$, and $k=-4$ gives $f(x)=(x-1)^{2}(x-4)$ with roots $1$ and $4$; either leaves exactly two distinct real roots.

# 6. Exercises

Give $\delta$, the stationary points and their values, and justify each count. Do not solve the cubics.

::::regions{tracks="1 1" gutter=18}
:::region
**A. Count the roots**

1. $x^{3}-12x+1$
2. $x^{3}+3x^{2}+3x+7$
3. $2x^{3}-9x^{2}+12x-5$
4. $-x^{3}+6x^{2}-9x+2$

**B. Find the parameter**

5. All $k$ for which $x^{3}-12x+k$ has exactly one real root.
6. All $k$ for which $x^{3}-3x^{2}+k$ has three distinct real roots.
7. All $a$ for which $x^{3}-3x^{2}+a$ has a repeated root, with the roots in each case.
:::
:::region
**C. Reason**

8. Show that $x^{3}+px+q$ has exactly one real root whenever $p\ge0$, for every $q$.
9. A classmate argues that two real zeros of $f'$ force $f$ to have three real roots. Give a counterexample and name the missing condition.
10. Explain why a cubic cannot have exactly two distinct real roots unless one of them is repeated.
:::
::::

::::keep
:::table{#tbl:record caption="Set out questions 1 to 4 under these headings before writing any conclusion"}
| Question | $\delta$ | Stationary points | Values | Product sign | Roots |
| ---: | --- | --- | --- | --- | ---: |
| 1 | | | | | |
| 2 | | | | | |
| 3 | | | | | |
| 4 | | | | | |
:::
::::

**Working space.**

________________________________________________________________________

________________________________________________________________________

## Sources

- Common Core State Standards, [HSA-APR.B.3](https://www.thecorestandards.org/Math/Content/HSA/APR/): identifying the zeros of polynomials.
- Common Core State Standards, [HSF-IF.C.7](https://www.thecorestandards.org/Math/Content/HSF/IF/): graphing polynomials and their end behaviour.
- OpenStax, [*Calculus Volume 1*](https://openstax.org/details/books/calculus-volume-1): the Intermediate Value Theorem and the first derivative test.

*The derivative test used here belongs to a first course in calculus and reaches past the two algebra standards cited; a class that has not met derivatives can replace Section 2 with the completed square of $f'$.*