Question

Given that one of the zeros of the cubic polynomial $a{{x}^{3}}+b{{x}^{2}}+d$ is zero, find the product of the other two zeros.

Answer 1

Hint: To find the product of the other two zeros, we will use the rule of sum and products of a cubic polynomial $a{{x}^{3}}+b{{x}^{2}}+cx+d=0$ . We know that the sum of product of the roots taken two at a time is $\dfrac{c}{a}$ , that is, $\alpha \beta +\beta \gamma +\gamma \alpha =\dfrac{c}{a}$ . According to the given condition, we will set $\alpha =0$ and substitute this in the equation formed in the previous step. Finally, we will substitute the value of c as 0 since the given cubic polynomial does not have coefficient of x.

Complete step by step solution:
We are given that one of the zeros of the cubic polynomial $a{{x}^{3}}+b{{x}^{2}}+d$ is zero. We have to find the product of the other two zeros. We know that for a quadratic equation $a{{x}^{3}}+b{{x}^{2}}+cx+d=0$ , the sum of the roots is $\dfrac{-b}{a}$ . Let $\alpha ,\beta ,\gamma $ be the roots of the cubic polynomial $a{{x}^{3}}+b{{x}^{2}}+cx+d=0$ .
$\Rightarrow \alpha +\beta +\gamma =\dfrac{-b}{a}$
We know that the product of the roots is $\dfrac{-d}{a}$ .
$\Rightarrow \alpha \beta \gamma =\dfrac{-d}{a}$
We also know that the sum of the product of the roots taken two at a time is $\dfrac{c}{a}$ .
$\Rightarrow \alpha \beta +\beta \gamma +\gamma \alpha =\dfrac{c}{a}...\left( i \right)$
Here, we will consider the equation (i) since we have to find the product of two other roots. We will equate $\alpha =0$ . Then, we can write the equation (i) as
$\begin{align}
  & \Rightarrow 0+\beta \gamma +0=\dfrac{c}{a} \\
 & \Rightarrow \beta \gamma =\dfrac{c}{a}...\left( ii \right) \\
\end{align}$
We obtained the product of the other two roots to be $\dfrac{c}{a}$ . Now, from the given cubic polynomial $a{{x}^{3}}+b{{x}^{2}}+d$ , we can see that $c=0$ . Let us substitute this in the equation (ii).
$\begin{align}
  & \Rightarrow \beta \gamma =\dfrac{0}{a} \\
 & \Rightarrow \beta \gamma =0 \\
\end{align}$
Therefore, the product of the other two roots of the given cubic polynomial is 0.

Note: Students should never forget to substitute $c=0$ at the end since the given cubic polynomial does not have a coefficient of x. They can also consider any of the other roots as 0 instead of $\alpha $ . Students must thoroughly understand and learn the properties of the roots, their sum and product.