Nature of Roots of Cubic Polynomial

Nature of Roots of Cubic Polynomial

A cubic polynomial is a polynomial of degree three, often seen in real-life applications such as calculating profit curves, population growth, or motion under varying acceleration. The nature of roots of a cubic polynomial helps determine whether it has one real root or three real roots (which may be real or complex).

The general form of a cubic polynomial is:
$f(x) = a x^3 + b x^2 + c x + d$

The corresponding cubic equation is:
$f(x) = 0$

where $a, b, c, d \in \mathbb{R}$ and $a > 0$. Since the degree is 3, there will always be three roots, real or complex.

Derivative and Discriminant of Cubic Polynomial

The first derivative of the function is given by:
$f'(x) = 3a x^2 + 2b x + c$

The discriminant of the derivative helps us understand the number of turning points of the cubic function. It is given by:
$D = 4a^2 - 12b = 4(a^2 - 3b)$

The discriminant ($D$) determines whether the cubic polynomial is strictly increasing, or has both a local maximum and minimum, which directly affects the nature of its roots.

Case 1: When $D < 0$

If $D < 0 \Rightarrow f'(x) > 0$ for all $x \in \mathbb{R}$,
then $f(x)$ is a strictly increasing function.

Also,
$\lim_{x \to -\infty} f(x) = -\infty$ and $\lim_{x \to \infty} f(x) = \infty$

From the graph, it is clear that $f(x)$ cuts the $x$-axis only once, implying one real root and two complex conjugate roots.

Clearly,
$x_0 > 0$ if $d < 0$, and $x_0 < 0$ if $d > 0$.

Case 2: When $D > 0$

If $D > 0$, then $f'(x) = 0$ has two distinct real roots, say $x_1$ and $x_2$, with $x_1 < x_2$.

Hence,
$f'(x) = 3a(x - x_1)(x - x_2)$

So,

$
f'(x) =
\begin{cases}
f'(x) > 0, & x \in (-\infty, x_1) \cup (x_2, \infty) \\
f'(x) = 0, & x = x_1 \text{ or } x_2 \\
f'(x) < 0, & x \in (x_1, x_2)
\end{cases}
$

Here, $x = x_1$ is the point of local maxima, and $x = x_2$ is the point of local minima.
Thus, the cubic polynomial cuts the $x$-axis at three distinct real points, meaning three real and unequal roots.

Case 3: When $D = 0$

If $D = 0$, then
$f'(x) = 3a(x - x_1)^2$

Here, $x_1$ is a repeated root of $f'(x) = 0$.
Hence,
$f(x) = a(x - x_1)^3 + C$

• If $C = 0$, then $f(x) = a(x - x_1)^3$ has three equal real roots.

• If $C \neq 0$, then $f(x) = 0$ has one real root and two complex conjugate roots.

Nature of Roots

Key Note:

• The discriminant of the derivative determines the nature of the roots of a cubic polynomial.

• When $D > 0$: three real and unequal roots.

• When $D = 0$: at least two equal real roots.

• When $D < 0$: one real and two complex conjugate roots.

Thus, the graph of y = f(x) could have five possibilities as shown below:

(i)

(ii)

(iii)

(iv)

(v)

Solved Examples Based on the Nature of Roots of Cubic Polynomial:

Example 1: Let a be an integer such that all real roots the polynomial $2 x^2+5 x^4+10 x^3+10 x^2+10 x+10$ lie in the interval $(\mathrm{a}, \mathrm{a}+1)$. Then, $|a|$ is equal to

1) 2

2) 4

3) 6

4) 5

Solution
$
\begin{aligned}
& \text { Let } 2 x^5+5 x^4+10 x^3+10 x^2+10 x+10=f(x) \\
& f(x)=x^5+x^5+5 x^4+10 x^3+10 x^2+5 x+10+5 x+9 \\
& f(x)=x^5+5 x+9+(x+1)^5 \\
& f^{\prime}(x)=5 x^4+5+0+5(x+1)^4>0
\end{aligned}
$

$\mathrm{f}(\mathrm{x})$ is an increasing function
Now $f(-2)=-34$ and $f(-1)=3$
Hence $f(x)$ has a root in $(-2,-1)$

So, $a=-2,|a|=2$.

Hence, the answer is the option 1.

Example 2: The number of distinct real roots of $x^4-4 x+1=0$ is :
1) 4
2) 2
3) 1
4) 0

Solution
$f(x)=x^4-4 x+1$
$f^{\prime}(x)=4 x^3-4=0$
$\Rightarrow \mathrm{x}=1$
$f^{\prime \prime}(x)=12 x^2, \quad f^{\prime \prime}(1): 12>0$
$\Rightarrow \mathrm{x}=1$ is a point of minima
$f(1): 1-4+1=-2$
For $\mathrm{x}<1, \quad \mathrm{f}(\mathrm{n})$ is decreasing and for $\mathrm{x}>1, \quad \mathrm{f}(\mathrm{x})$ is in increasing

$
\mathrm{f}(-\infty)=\infty, \quad \mathrm{f}(1)=-2, \quad \mathrm{f}(\infty)=\infty
$
From the intermediate value theorem, $f(x)$ will have 2 real roots one less than 1 and the other greater than 1.
Hence, the answer is the option (2).

Example 3: Let $\alpha, \beta, \gamma$, be the three roots of the equation $x^3+b x+c=0$ If $\beta \gamma=1=-\alpha$, then $b^3+2 c^3-3 \alpha^3-6 \beta^3-8 \gamma^3$ is
1) $\frac{155}{8}$
2) 21
3) 19
4) $\frac{169}{8}$

Solution

$
\begin{aligned}
& \beta \gamma=1 \\
& \alpha=-1 \\
& \text { Put } \alpha=-1 \\
& -1-b+c=0 \\
& c-b=1
\end{aligned}
$

also

$
\begin{aligned}
& \alpha \cdot \beta \cdot \gamma=-\mathrm{c} \\
& -1=-\mathrm{c} \Rightarrow \mathrm{c}=1 \\
& \therefore \mathrm{b}=0 \\
& \mathrm{x}^3+1=0 \\
& \alpha=-1, \beta=-\mathrm{w}, \gamma=-\mathrm{w}^2 \\
& \therefore \mathrm{b}^3+2 \mathrm{c}^3-3 \alpha^3-6 \beta^3-8 \gamma^3 \\
& 0+2+3+6+8=19
\end{aligned}
$

List of topics related to Nature of roots of cubic polynomial

This section covers all the interconnected concepts that help understand the behavior of cubic functions, including their continuity, differentiability, and graphical interpretation. It’s useful for building a clear foundation before studying the nature of roots in detail.

Differentiability and Existence of Derivative

Examining differentiability Using Graph of Function

Continuity of Composite Function

Continuity And Differentiability

NCERT Resources

This section lists NCERT Class 12 Maths study materials that support the topic. These include notes, textbook solutions, and exemplar questions for the “Continuity and Differentiability” chapter, helping you prepare systematically.

NCERT Class 12 Maths Notes for Chapter 5 - Continuity and Differentiability

NCERT Class 12 Maths Solutions for Chapter 5 - Continuity and Differentiability

NCERT Class 12 Maths Exemplar Solutions for Chapter 5 - Continuity and Differentiability

Practice Questions based on Nature of roots of cubic polynomial

This segment provides practice questions designed to test your understanding of cubic polynomial roots, differentiability, and graphical properties. Each question strengthens conceptual and exam-level problem-solving skills.

Nature Of Roots Of Cubic Polynomial- Practice Question MCQ

We have shared below the links to practice questions on the related topics to Nature of roots of cubic polynomial: