Question

If the sum of zeroes of the polynomial \[f\left( x \right) = 2{x^3} - 3k{x^2} + 4x - 5\] is 6, then the value of \[k\] is
(a) 2 (b) 4 (c) \[ - 2\] (d) \[ - 4\]

Answer 1

Hint:
Here, we need to find the value of \[k\]. We need to use the relation of the zeroes and coefficients of the variables of the cubic polynomial to obtain an equation in terms of \[k\]. We will solve this equation to find the required value of \[k\].

Formula used:
The sum of zeroes of a cubic polynomial is equal to the negative ratio of the coefficient of \[{x^2}\], and the coefficient of \[{x^3}\], that is \[\alpha + \beta + \gamma = - \dfrac{b}{a}\].

Complete step by step solution:
The given polynomial is a cubic polynomial.
Rewriting the equation, we get
\[f\left( x \right) = 2{x^3} - 3k{x^2} + 4x - 5 = 0\]
We will use the relation of the zeroes and coefficients of the variables of the cubic polynomial.
The sum of zeroes of a cubic polynomial is equal to the negative ratio of the coefficient of \[{x^2}\], and the coefficient of \[{x^3}\], that is \[\alpha + \beta + \gamma = - \dfrac{b}{a}\].
Comparing the coefficients of the standard form of a cubic polynomial \[a{x^3} + b{x^2} + cx + d = 0\], and the equation \[2{x^3} - 3k{x^2} + 4x - 5 = 0\], we get
\[a = 2\] and \[b = - 3k\]
Substituting \[a = 2\] and \[b = - 3k\] in the formula for sum of zeroes of a cubic polynomial, we get
\[ \Rightarrow \alpha + \beta + \gamma = - \dfrac{{\left( { - 3k} \right)}}{2}\]
Simplifying the expression, we get
\[ \Rightarrow \alpha + \beta + \gamma = \dfrac{{3k}}{2}\]
It is given that the sum of zeroes of the given cubic polynomial is 6.
Therefore, we get
\[ \Rightarrow \alpha + \beta + \gamma = 6\]
From the equations \[\alpha + \beta + \gamma = \dfrac{{3k}}{2}\] and \[\alpha + \beta + \gamma = 6\], we get the equation
\[ \Rightarrow \dfrac{{3k}}{2} = 6\]
This is a linear equation in terms of \[k\]. We will solve this equation to find the value of \[k\].
Multiplying both sides by 2, we get
\[ \Rightarrow 3k = 12\]
Dividing both sides by 3, we get
\[ \Rightarrow k = 4\]
\[\therefore \] We get the value of \[k\] as 4.

Thus, the correct option is option (b).

Note:
We have formed a linear equation in one variable in terms of \[k\] in the solution. A linear equation in one variable is an equation of the form \[ax + b = 0\], where \[b\] and \[a\] are integers and have only one solution.
The given equation is a cubic equation. A cubic equation is an equation of the form \[a{x^3} + b{x^2} + cx + d = 0\], where \[a\] is not equal to 0, and \[d\] is a constant and also the highest degree of variable is 3.