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Definition of Function

If x-c is a...

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 a root of the polynomial $($ where $f^{'} (x)$ represents $r^{t h}$ derivative of $f (x)$ w.r.t. $x)$

f^{m} (x)

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f^{m - 1} (x)

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f^{n} (x)

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none of these

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Solution

The correct option is B $f^{m - 1} (x)$
From the given information, we have
$f (x) = {(x - c)}^{m} g (x),$
where $g (x)$ is a polynomial of degree $n - m$
Then $x = c$ is a common root for equation $f (x) = 0, f^{1} (x) = 0, f^{2} (x) = 0, . . . . . . . = f^{m - 1} (x) = 0$

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If x−c is a factor of order m of the polynomial f(x) of degree n(1<m<n), then x=c is a root of the polynomial ( where f′(x) represents rth derivative of f(x) w.r.t. x)

The correct option is B fm−1(x)From the given information, we havef(x)=(x−c)mg(x),where g(x) is a polynomial of degree n−mThen x=c is a common root for equation f(x)=0,f1(x)=0,f2(x)=0,.......=fm−1(x)=0

If $x - c$ is a factor of order $m$ of the polynomial $f (x)$ of degree $n (1 < m < n)$ , then $x = c$ is a root of the polynomial $($ where $f^{'} (x)$ represents $r^{t h}$ derivative of $f (x)$ w.r.t. $x)$

The correct option is B $f^{m - 1} (x)$
From the given information, we have
$f (x) = {(x - c)}^{m} g (x),$
where $g (x)$ is a polynomial of degree $n - m$
Then $x = c$ is a common root for equation $f (x) = 0, f^{1} (x) = 0, f^{2} (x) = 0, . . . . . . . = f^{m - 1} (x) = 0$