Question

If $f\left(x\right)=2x^3+m{x}^2-13x+n$ and $2,3$ are the roots of the equation $f\left(x\right)=0$, then the values of $m$ and $n$ are:

• A. $-5,-30$
• B. $-5,30$
• C. $5,30$
• D. None of these

Answer 1

The correct option is B

$-5,30$

Explanation for the correct option.

Step 1: Find the value of the given function for all the given roots.

The two roots of the cubic equation are given to be $2\mathrm{and}3$.

So, for the given equation,

$$\begin{gathered} f\left(2\right)=2{\left(2\right)}^3+m{\left(2\right)}^2-13\left(2\right)+n=0 \\ \Rightarrow 16+4m-26+n=0 \\ \Rightarrow 4m+n=10.....\left(1\right) \end{gathered}$$

Similarly,

$$\begin{gathered} f\left(3\right)=2{\left(3\right)}^3+m{\left(3\right)}^2-13\left(3\right)+n=0 \\ \Rightarrow 54+9m-39+n=0 \\ \Rightarrow 9m+n=-15......\left(2\right) \end{gathered}$$

Step 2: Find the value of $m$ and $n$.

By subtracting $\left(1\right)$ from $\left(2\right)$, we get

$\begin{array}{rcl}5m& =& -25\\ & \Rightarrow & m=-5\end{array}$

Substituting the value of $m$ in $\left(1\right)$, we get

$\begin{array}{rcl}4\left(-5\right)+n& =& 10\\ \Rightarrow n& =& 30\end{array}$

Therefore, $m=-5\mathrm{and}n=30$.

Hence, option B is correct.