Question

If the sum of the zeroes of the polynomial $ f(x) = 2{x^3} - 3k{x^2} + 4x - 5 $ is $ 6 $ , then the value of $ k $ is:
A. $ 2 $
B. $ 4 $
C. $ - 2 $
D. $ - 4 $

Answer 1

Hint:
Here we just need to know that if we are given the cubic equation of the form $ a{x^3} + b{x^2} + cx + d = 0 $ then the sum of the roots is given by the formula $ - \dfrac{b}{a} $ and so we can find the sum of the roots by the given cubic equation and equate it to $ 6 $ and get the required value.

Complete step by step solution:
Here we are given the cubic equation $ f(x) = 2{x^3} - 3k{x^2} + 4x - 5 $ and we are given that the zeroes of this polynomial have the sum as $ 6 $
We know that zeroes of the polynomial means the values that make the polynomial equate to zero. The values of the variable at which the given function becomes zero is called the zeroes of the given polynomial. Zeroes of the polynomial are actually the roots of the polynomial because at these values of the variable the polynomial becomes zero. So we are given that sum of the roots of the cubic equation $ 2{x^3} - 3k{x^2} + 4x - 5 = 0 $ is $ 6 $
We know that if we are given the cubic equation which is represented by the form $ a{x^3} + b{x^2} + cx + d = 0 $ then the sum of the roots is given by the formula $ - \dfrac{b}{a} = - \dfrac{{{\text{coefficient of }}{x^2}}}{{{\text{coefficient of }}{x^3}{\text{ }}}} $ and so we can find the sum of the roots by the given cubic equation and equate it to $ 6 $ and get the required value.
So we have the equation $ 2{x^3} - 3k{x^2} + 4x - 5 = 0 $
The coefficient of $ {x^2} = - 3k $
Coefficient of $ {x^3} = 2 $
So we can say the sum of roots $ - \dfrac{{{\text{coefficient of }}{x^2}}}{{{\text{coefficient of }}{x^3}{\text{ }}}} = - \dfrac{{ - 3k}}{2} = \dfrac{{3k}}{2} $
We are given that sum of roots has the value $ 6 $
So we can write that
 $
  \dfrac{{3k}}{2} = 6 \\
  k = \dfrac{{2(6)}}{3} = \dfrac{{12}}{3} = 4 \\
  $

Hence, we can say that B is the correct option.

Note:
Here we must all know the properties of the cubic function like:
If we have the equation $ a{x^3} + b{x^2} + cx + d = 0 $ then:
Sum of roots $ = - \dfrac{b}{a} $
Product of all roots $ = \dfrac{{ - d}}{a} $
Sum of product of the roots taken two at a time $ = \dfrac{c}{a} $
These formulas make the problem simpler and easier to solve.