Question

If (x-c) is a factor of order m of the polynomial f(x) of degree n(1 < m < n), then x=c is the root of the polynomial(where ${{f}^{r}}\left( x \right)$ represents ${{r}^{th}}$ derivative of f(x) w.r.t. x).

(a) ${{f}^{m}}\left( x \right)$
(b) ${{f}^{m-1}}\left( x \right)$
(c) ${{f}^{n}}\left( x \right)$
(d) none of these

Answer 1

First, before proceeding for this, we must suppose a polynomial g(x) with degree (n-m) that is multiplied with the factor (x-c) with degree m to get the another polynomial which is defined in the question as f(x) as $f\left( x \right)={{\left( x-c \right)}^{m}}g\left( x \right)$ . Then, x=c is a common root for all equations that comes from the derivative of the above function. Then, if we differentiate the above expression to the last degree to get (x=c) as a root, we get the last differentiation as ${{f}^{m-1}}\left( x \right)=0$ which gives the final result.

Complete step-by-step answer:
In this question, we are supposed to find if x=c is the root of the polynomial when (x-c) is a factor of order m of the polynomial f(x) of degree n(1So, before proceeding for this, we must suppose a polynomial g(x) with degree (n-m) that is multiplied with the factor (x-c) with degree m to get the other polynomial which is defined in the question as f(x).
$f\left( x \right)={{\left( x-c \right)}^{m}}g\left( x \right)$
Then, x=c is a common root for all equation that comes from the derivative of the above function as:
$f\left( x \right)=0,{f}'\left( x \right)=0,{f}''\left( x \right)=0....$ so on till the last term before the degree gets totally eliminated as ${{f}^{m-1}}\left( x \right)=0$ .
So, ${{f}^{r}}\left( x \right)$ represents the ${{r}^{th}}$ derivative of the polynomial function f(x).
Moreover, if we differentiate the above expression to the last degree to get (x=c) as a root, we get the last differentiation as ${{f}^{m-1}}\left( x \right)=0$ .
So, the (x=c) is the root of the polynomial ${{f}^{m-1}}\left( x \right)$ .

So, the correct answer is “Option b”.

Note: Now, to solve these type of the questions we need to know some of the basics of the differentiation as let us suppose the polynomial as $f\left( x \right)={{x}^{n}}$ . Then, if we need to eliminate the entire power of the polynomial with power n, we need to perform the differentiation (n-1) times. So, the factor (x=c ) only occurs when ${{f}^{n-1}}\left( x \right)=0$ is calculated.