Question

If ${{x}^{3}}+m{{x}^{2}}-x+6$ has $\left( x-2 \right)$ as a factor and leaves remainder “n” when divided by $\left( x-3 \right)$, find the value of m and n?

Answer 1

Hint: It is given that $\left( x-2 \right)$ is a factor of ${{x}^{3}}+m{{x}^{2}}-x+6$ so putting the value of x as 2 in this expression and equate this expression to 0. And in this way, we can find the value of m. Now, to find the value of n, we are going to put the value of m calculated in the above and then substitute the value of x as 3 in this expression and solve the expression. The evaluation of this expression will give the value of remainder (or n).

Complete step by step answer:
The polynomial expression given in the above problem is as follows:
${{x}^{3}}+m{{x}^{2}}-x+6$
As $\left( x-2 \right)$ is a factor of the above expression, so putting the value of x as 2 will make the above expression equal to 0 according to the factor theorem.
$\begin{align}
  & {{\left( 2 \right)}^{3}}+m{{\left( 2 \right)}^{2}}-2+6 \\
 & \Rightarrow 8+4m-4 \\
 & \Rightarrow 4m+4 \\
\end{align}$
Equating the above expression to 0 we get,
$4m+4=0$
Subtracting 4 on both the sides we get,
$4m=-4$
Dividing 4 on both the sides of the above equation and we get,
$m=-1$
From the above, we have solved the value of m as -1
Now, substituting this value of m in the above polynomial expression in “n” and we get,
$\begin{align}
  & {{x}^{3}}+\left( -1 \right){{x}^{2}}-x+6 \\
 & \Rightarrow {{x}^{3}}-{{x}^{2}}-x+6 \\
\end{align}$
Now, we are asked to find the remainder by dividing $\left( x-3 \right)$ to the above polynomial expression which we are going to find by using the remainder theorem in which we are going to put x as 3 in the above expression and then see what value of the expression we are getting.
$\begin{align}
  & {{\left( 3 \right)}^{3}}-{{3}^{2}}-3+6 \\
 & =27-9-3+6 \\
 & \Rightarrow 33-12 \\
 & =21 \\
\end{align}$
From the above, we got the value of remainder as 21 or the value of n as 21.

Hence, we have found the value of m and n as -1 and 21 respectively.

Note: To solve the above problem, we need to know about factor theorem and the remainder theorem otherwise you cannot get the value of “m” and hence, will not get the value of n.
You might be thinking that instead of using the remainder theorem while finding the value of n, you can divide $\left( x-3 \right)$ by the given polynomial so the answer is yes, you can do it but it will take a lot of time so in the competitive exams you can use the remainder theorem to quickly get the answer.