IIT JEE 1983 Maths - True or False Q3

Calculus Level 3

True or False?

If x − r x-r is a factor of the polynomial f ( x ) = a n x n + ⋯ + a 1 x + a 0 f(x) = a_n x^n + \cdots + a_1 x + a_0 repeated m m times, with 1 < m ≤ n 1< m \leq n , then r r is a root of f ′ ( x ) = 0 f'(x)=0 repeated m m times.

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False True

Shubhamkar Ayare by Shubhamkar Ayare , 22, India

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1 solution

Chew-Seong Cheong
Jan 11, 2017

f ( x ) = a n x n + . . . + a 2 x 2 + a 1 x + a 0 can be written as = ( x − r ) m g ( x ) where g ( x ) has a degree of n − m . ⟹ f ′ ( x ) = m ( x − r ) m − 1 g ( x ) + ( x − r ) m g ′ ( x ) = ( x − r ) m − 1 [ m g ( x ) + ( x − r ) g ′ ( x ) ] \begin{aligned} f(x) & = a_nx^n + ... + a_2x^2 + a_1x + a_0 & \small \color{#3D99F6} \text{can be written as} \\ & = (x-r)^m \color{#3D99F6} g(x) & \small \color{#3D99F6} \text{where }g(x) \text{ has a degree of }n-m. \\ \implies f'(x) & = m(x-r)^{m-1}g(x) + (x-r)^m g'(x) \\ & = (x-r)^{m-1}[mg(x) + (x-r)g'(x)] \end{aligned}

When f ′ ( x ) = 0 f'(x) = 0 , r r is a root but only repeated m − 1 m-1 times. Therefore, the answer is False \boxed{\text{False}} .

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