Question

If the zeroes of the cubic polynomial $\mathrm{f}\left(\mathrm{x}\right)={\mathrm{kx}}^3-8{\mathrm{x}}^2+5$ are $\mathrm{\alpha }-\mathrm{\beta }$, $\mathrm{\alpha }$ and $\mathrm{\alpha }+\mathrm{\beta }$ then find the value of $\mathrm{k}$ .

Answer 1

Find the value of $\mathrm{k}$ .

As we know that, The cubic polynomial is in the form ${\mathbf{ax}}^{\mathbf{3}\mathbf{}}\mathbf{+}\mathbf{}{\mathbf{bx}}^{\mathbf{2}}\mathbf{}\mathbf{+}\mathbf{}\mathbf{cx}\mathbf{}\mathbf{+}\mathbf{}\mathbf{d}\mathbf{}$and the zeroes are $\mathrm{\alpha },\mathrm{\beta },$ and $\mathrm{\gamma }$.

It is given that the polynomial cubic equation is $\mathrm{f}\left(\mathrm{x}\right)={\mathrm{kx}}^3-8{\mathrm{x}}^2+5$.

Here,

$\mathrm{a}=\mathrm{k}$

$\mathrm{b}=-8$

$\mathrm{c}=0$

$\mathrm{d}=5$

We will find the value of $\mathrm{k}$ by using the sum of roots of cubic polynomial.

The sum of the roots is,

$\mathbf{\alpha }\mathbf{+}\mathbf{\beta }\mathbf{=}\mathbf{-}\frac{\mathbf{b}}{\mathbf{a}}$

Where, $\mathrm{b}$ is the coefficient of ${\mathrm{x}}^2$ and $\mathrm{a}$ is the coefficient of ${\mathrm{x}}^3$ .

The roots of the cubic polynomial are $\left(\mathrm{\alpha }-\mathrm{\beta }\right),\left(\mathrm{\alpha }\right)$ and $\left(\mathrm{\alpha }+\mathrm{\beta }\right)$.

Sum of roots is,

$\left(\mathrm{\alpha }-\mathrm{\beta }\right)+\left(\mathrm{\alpha }\right)+\left(\mathrm{\alpha }+\mathrm{\beta }\right)=3\mathrm{\alpha }$ -------- $\left(1\right)$

Substitute the given values in the formula $\mathbf{-}\frac{\mathbf{b}}{\mathbf{a}}$.

$\Rightarrow$ $\left(\frac{-\mathrm{b}}{\mathrm{a}}\right)=-\left(\frac{-8}{\mathrm{k}}\right)$ -------- $\left(2\right)$

Equate both equations $\left(1\right)$ and $\left(2\right)$.

$\Rightarrow$ $3\mathrm{\alpha }=-\left(\frac{-8}{\mathrm{k}}\right)$

$\Rightarrow$ $3\mathrm{\alpha }=\frac{8}{\mathrm{k}}$

$\Rightarrow$ $\mathrm{k}=\frac{8}{3\mathrm{\alpha }}$

Hence the value of $\mathrm{k}$ is $\frac{8}{3\mathrm{\alpha }}$ for the cubic polynomial equation $\mathrm{f}\left(\mathrm{x}\right)={\mathrm{kx}}^3-8{\mathrm{x}}^2+5$.