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,
$\text { Sum of product of two zeroes at a time }=\frac{c}{a}$
$\Rightarrow \alpha \beta+\beta \gamma+\gamma \alpha=\frac{c}{a}$
$\Rightarrow 0 \times \beta+\beta \gamma+\gamma \times 0=\frac{c}{a}[\because \beta=0, \text { given }]$
$\Rightarrow 0+\beta \gamma+0=\frac{c}{a}$
$\Rightarrow \beta \gamma=\frac{c}{a}$
Hence, Product of other two zeroes $=\frac{ c }{ a }$.

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