Question 5

If one of the zeroes of the cubic polynomial $a{x}^3+b{x}^2+cx+d$ is zero, then the product of other two zeroes is

Solution:

Let $p\left(x\right)=a{x}^3+b{x}^2+cx+d$
Given that, one of the zeroes of the cubic polynomial $p\left(x\right)$ is zero.
Let $\alpha ,\beta$ and $\gamma$ are the zeroes of cubic polynomial $p\left(x\right)$, where $a=0$. We known that,
Sum of product of two zeroes at a time $=\frac{\mathrm{c/}}{a}$
$\Rightarrow \alpha \beta +\beta \gamma +\gamma \alpha =\frac{\mathrm{c/}}{a}$
$\Rightarrow 0\times \beta +\beta \gamma +\gamma \times 0=\frac{\mathrm{c/}}{a}[\because \alpha =0,\text{given}]$
$\Rightarrow 0+\beta \gamma +0=\frac{\mathrm{c/}}{a}$
$\Rightarrow \beta \gamma =\frac{\mathrm{c/}}{a}$
Hence, product of other two zeroes $=\frac{\mathrm{c/}}{a}$