Question

If 3 is a zero of the polynomial $f\left( x \right)={{x}^{4}}-{{x}^{3}}-8{{x}^{2}}+kx+12$, then the value of k is,
A. – 2
B. 2
C. – 3
D. $\dfrac{3}{2}$

Answer 1

Hint: We will start by using the fact that the zero of a polynomial $f\left( x \right)$ is the value of x for which $f\left( x \right)=0$. Then we will use this fact to find the value of k by using substituting x = 3 in $f\left( x \right)$ and equating it to zero.

Complete step-by-step answer:
Now, we have been given a polynomial as $f\left( x \right)={{x}^{4}}-{{x}^{3}}-8{{x}^{2}}+kx+12$. Now, we will use the fact that the zero of a polynomial $f\left( x \right)$ is the value of x for which the function $f\left( x \right)=0$. Therefore, using the same concept the value of $f\left( x \right)$ at x = 3 must be zero. Therefore, we have,
$f\left( 3 \right)=0$
Now, we have $f\left( x \right)={{x}^{4}}-{{x}^{3}}-8{{x}^{2}}+kx+12$
$\begin{align}
  & \Rightarrow {{3}^{4}}-{{3}^{3}}-8\times {{3}^{2}}+k\times 3+12=0 \\
 & \Rightarrow 81-27-8\times 9+3k+12=0 \\
 & \Rightarrow 81-27-72+3k+12=0 \\
 & \Rightarrow 81+12-27-72+3k=0 \\
 & \Rightarrow 93-99+3k=0 \\
 & \Rightarrow -6+3k=0 \\
 & \Rightarrow 3k=6 \\
 & \Rightarrow k=2 \\
\end{align}$
Hence, the value of k is 2 and the correct option is (B).

Note: To solve this question we have used a fact that the zero of a polynomial is the value at which the function is zero while solving the equation by substituting the value of zero in $f\left( x \right)$. It is important to do calculations carefully to avoid mistakes.