Nature of Roots of Cubic Polynomial - Practice Questions & MCQ

Nature of Roots of Cubic Polynomial

Let the cubic polynomial be $f(x)= an x^3+b x^2+c x+d$ and $f(x)=0$ is a cubic equation where $a, b, c$ and $d \in R$ and $a>0$.
Now, $f^{\prime}(x)=3 a x^2+b x+c$
Now, $\quad \mathrm{f}^{\prime}(\mathrm{x})=\mathrm{ax}^2+\mathrm{bx}+\mathrm{c}$
Let $D=4 a^2-12 b=4\left(a^2-3 b\right)$ be the discriminant of the equation $f^{\prime}(x)=0$
Now, we will have the following cases
case 1
If $\mathrm{D}<0 \Rightarrow \mathrm{f}^{\prime}(\mathrm{x})>0 \forall \mathrm{x} \in \mathrm{R}$.
That means $f(x)$ would be an increasing function of x
Also, $\lim _{x \rightarrow-\infty} f(x)=-\infty$ and $\lim _{x \rightarrow \infty} f(x)=\infty$
Also, from the graph, it is clear that $\mathrm{f}(\mathrm{x})$ cut the $\mathrm{x}-$ axis only once.
Clearly $x_0>0$ if $d<0$, and $x_0<0$ if $d>0$
case 2
If $\mathrm{D}>0 \Rightarrow \mathrm{f}^{\prime}(\mathrm{x})=0$ would have two real roots, say $\mathrm{x}_1$ and $\mathrm{x}_2$
let $\mathrm{x}_1<\mathrm{x}_2$

$
\begin{array}{lrl}
\Rightarrow & f^{\prime}(x)=3 a\left(x-x_1\right)\left(x-x_2\right) \\
\Rightarrow & f^{\prime}(x)= \begin{cases}f^{\prime}(x)<0, & x \in\left(x_1, x_2\right) \\
f^{\prime}(x)=0, & x \in\left\{x_1, x_2\right\} \\
f^{\prime}(x)>0 & \left(-\infty, x_1\right) \cup\left(x_2, \infty\right)\end{cases}
\end{array}
$
Here, $\mathrm{x}=\mathrm{x}_1$ is point of local maxima and $\mathrm{x}=\mathrm{x}_2$ is point of local minima

case 3
If $\mathrm{D}=0 \Rightarrow \mathrm{f}^{\prime}(\mathrm{x})=3 \mathrm{a}\left(\mathrm{x}-\mathrm{x}_1\right)^2$
When, $x_1$ is root of $f^{\prime}(x)=0$, then $f(x)=a\left(x-x_1\right)^3+C$.
If $\mathrm{C}=0$, then $\mathrm{f}(\mathrm{x})=\mathrm{a}\left(\mathrm{x}-\mathrm{x}_1\right)^3$ has 3 equal roots
if, $\mathrm{C} \neq 0$, then $\mathrm{f}(\mathrm{x})=0$ has one real root.
Thus, the graph of $y=f(x)$ could have five possibilities as shown below:

(i)

(ii)

(iii)

(iv)

(v)

Conclusion:

a. If $f\left(x_1\right) f\left(x_2\right)>0, f(x)=0$ would have just one real root.
b. If $f\left(x_1\right) f\left(x_2\right)<0, f(x)=0$ would have three real and distinct roots.
c. If $f\left(x_1\right) f\left(x_2\right)=0, f(x)=0$ would have three real roots but one of the roots would be repeated.