Question

If one of the zeroes of the cubic polynomial \[a{{x}^{3}}+b{{x}^{2}}+cx+d\] is zero, the product of the other two zeroes is
A.$\dfrac{-c}{a}$
B.$\dfrac{c}{a}$
C.0
D.$\dfrac{-2}{3}$

Answer 1

Hint: For solving this problem, first we satisfied the given zero in the cubic polynomial. By doing so we obtain d = 0. By this our polynomial is reduced to \[a{{x}^{3}}+b{{x}^{2}}+cx\]. Now, by using the general form, we express the sum and product of roots for a cubic polynomial. By putting one of the roots as zero we obtain the product of the other two roots.

Complete Step-by-step answer:
The given cubic equations, \[a{{x}^{3}}+b{{x}^{2}}+cx+d=0\].
Also, we know that 0 is one of the zeroes of the cubic polynomial equation.
Putting the value of x as 0 in the cubic polynomial, we get
$\begin{align}
  & a\times 0+b\times 0+c\times 0+d=0 \\
 & \therefore d=0 \\
\end{align}$
Hence, the simplified cubic polynomial is \[a{{x}^{3}}+b{{x}^{2}}+cx\].
Now, a cubic function is a polynomial function with one or more variables in which the highest-degree term is of the third degree. A single-variable cubic function can be stated as:
$f(x)=a{{x}^{3}}+b{{x}^{2}}+cx+d,\quad a\ne 0$
Let $\alpha ,\beta \text{ and }\gamma $ are zeroes of the cubic polynomial equation. Therefore, three properties associated with roots can be illustrated as:
$\alpha +\beta +\gamma =\dfrac{-b}{a}$
$\alpha \beta +\beta \gamma +\alpha \gamma =\dfrac{c}{a}\ldots (1)$
\[\alpha \beta \gamma =\dfrac{d}{a}\]
Now, for our cubic equation \[a{{x}^{3}}+b{{x}^{2}}+cx\], by using the equation (1) and satisfying $\gamma =0$, we get
$\begin{align}
  & \alpha \beta +\beta \gamma +\alpha \gamma =\dfrac{c}{a},\gamma =0 \\
 & \therefore \alpha \beta =\dfrac{c}{a} \\
\end{align}$
Thus, the product of the other two zeroes is $\alpha \beta =\dfrac{c}{a}$.
Hence, option (b) is correct.

Note: Students must remember the expression for sum and product of roots associated with cubic polynomials for solving this problem. The key steps involved in solving this problem is one of the roots is zero. So, by using individual products of two roots, we obtained the answer.