Choose the correct answer from the given four options in the foll… - Vidyadip

MCQ

Choose the correct answer from the given four options in the following questions : Given that one of the zeroes of the cubic polynomial $a x^3+b x^2+c x+d$ is zero, the product of the other two zeroes is :

A
$-\frac{\text{c}}{\text{a}}$
✓
$\frac{\text{c}}{\text{a}}$
C
$0 $
D
$-\frac{\text{b}}{\text{a}}$

✓

Answer

Correct option: B.

$\frac{\text{c}}{\text{a}}$

Let $p(x)=a x^3+b x^2+c x+d$
Given that, one of the zeroes of the cubic polynomial $p(x)$ is zero,
Let $\alpha,$
$\beta$ and $\gamma$ are the zeroes of cubic polynomial $p(x),$ where $a = 0.$
We know that,
Sum of product of two zeroes at a time $=\frac{\text{c}}{\text{a}}$
$\Rightarrow\ \alpha\beta + \beta\gamma + \gamma\alpha =\frac{\text{c}}{\text{a}}$
$\Rightarrow\ 0\times\beta+\beta\gamma +\gamma\times0=\frac{\text{c}}{\text{a}}\ \big[\because\ \beta = 0,$ given $\big]$
$\Rightarrow\ 0+\beta\gamma + 0 =\frac{\text{c}}{\text{a}}$
$\Rightarrow\ \beta\gamma=\frac{\text{c}}{\text{a}}$
Hence, Product of other two zeroes $=\frac{\text{c}}{\text{a}}.$

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