Question

A cubic polynomial with sum of its zeroes, sum of the product of its zeroes taken two at a time and the product of its zeroes as \[2, - 7, - 14\] respectively is
A. \[{x^3} - 2{x^2} - 7x - 14\]
B. \[{x^3} + 2{x^2} - 7x - 14\]
C. \[{x^3} + 2{x^2} + 7x - 14\]
D. \[{x^3} - 2{x^2} - 7x + 14\]

Answer 1

Hint: In this question, first of all consider the roots or zeros of the required cubic polynomial as variables. Then find the sum of the roots, products of the roots taken two at a time and the product of the roots by using the given data. Then substitute them in the cubic polynomial to get the final answer. So, use this concept to reach the solution of the given problem.

Complete step-by-step answer:
Let the zeros or roots of the given cubic polynomial are \[\alpha ,\beta ,\gamma \].
We know that the equation of a cubic polynomial with roots or zeroes as \[\alpha ,\beta ,\gamma \] is given by \[{x^3} - \left( {\alpha + \beta + \gamma } \right){x^2} + \left( {\alpha \beta + \alpha \gamma + \beta \gamma } \right)x - \alpha \beta \gamma \].
Hence, the equation of the given cubic polynomial with roots \[\alpha ,\beta ,\gamma \] is \[{x^3} - \left( {\alpha + \beta + \gamma } \right){x^2} + \left( {\alpha \beta + \alpha \gamma + \beta \gamma } \right)x - \alpha \beta \gamma \].
Given sum of roots i.e., \[\alpha + \beta + \gamma = 2\]
Sum of product of its zeroes taken two at a time i.e., \[\alpha \beta + \alpha \gamma + \beta \gamma = - 7\]
Product of its zeroes i.e., \[\alpha \beta \gamma = - 14\]
Substituting these values in the cubic polynomial, we get
\[
   \Rightarrow {x^3} - 2{x^2} + \left( { - 7} \right)x - \left( { - 14} \right) \\
  \therefore {x^3} - 2{x^2} - 7x + 14 \\
\]
Therefore, the required cubic polynomial is \[{x^3} - 2{x^2} - 7x + 14\].
Thus, the correct option is D. \[{x^3} - 2{x^2} - 7x + 14\]

Note:A cubic polynomial is a polynomial of degree 3. A cubic polynomial is of the form \[a{x^3} + b{x^2} + cx + d\]. An equation involving a cubic polynomial is called a cubic equation. A cubic equation is of the form \[a{x^3} + b{x^2} + cx + d = 0\].