Question

Given that 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) \[ - \dfrac{b}{a}\]

Answer 1

Hint:
Here we need to find the product of two zeroes of the given cubic polynomial. The zeroes of the polynomial means the roots of the polynomial. We will substitute the value of the given root in the cubic equation to get the value of constant of the cubic polynomial. Then we will put the value of constant in the equation. We will then get a quadratic equation. We will further solve the equation to find the zeros and multiply the obtained zeros to find the product.

Complete step by step solution:
We know zeroes of the polynomial means the roots of the polynomial.
It is given that 0 is the root of the given cubic polynomial, so it will satisfy the cubic equation.
Substituting the value of root in the cubic equation, \[a{x^3} + b{x^2} + cx + d\], we get
\[ \Rightarrow a{\left( 0 \right)^3} + b{\left( 0 \right)^2} + c\left( 0 \right) + d = 0\]
On further simplification, we get
\[ \Rightarrow d = 0\]
Thus, substituting the value of \[d\] , the cubic polynomial becomes
\[\begin{array}{l}a{x^3} + b{x^2} + cx + 0 = 0\\ \Rightarrow a{x^3} + b{x^2} + cx = 0\end{array}\]
We can write the equation as
\[ \Rightarrow x\left( {a{x^2} + bx + c} \right) = 0\]
Dividing the above equation by\[x\], we get
\[ \Rightarrow a{x^2} + bx + c = 0\]
The obtained equation is a quadratic equation and the product of roots of this equation is equal to the ratio of constant term i.e. \[c\] to the coefficient of \[{x^2}\].
Therefore, the product of roots of this quadratic equation is equal to \[\dfrac{c}{a}\].

Hence, the correct option is option B.

Note:
We need to keep in mind that the number of roots of the polynomial is always equal to the highest power in the polynomial. The number of roots of a quadratic polynomial is 2 because the highest power in a quadratic polynomial is 2. Similarly the number of roots of cubic polynomials is 3 because the highest power in cubic polynomials is 3.