But where is the polynomial?

Calculus Level 3

The sum of roots of the 57th derivative with respect to x x of a polynomial p ( x ) p(x) of degree 94 with a non-zero leading coefficient is 111. Then find the sum of roots of 88th derivative with respect to x x of the polynomial p ( x ) p(x) .


The answer is 18.

Tapas Mazumdar by Tapas Mazumdar , 20, India

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1 solution

Tapas Mazumdar
Oct 5, 2016

Refer to this note for details on my method of solving this problem.

By our 'mean value approach', we know that,

Mean of roots of 5 7 t h derivative = Mean of roots of 8 8 t h derivative \text{Mean of roots of} \ 57^{th} \ \text{derivative} = \text{Mean of roots of} \ 88^{th} \ \text{derivative}

⟹ Sum of roots of 5 7 t h derivative Total number of roots in 5 7 t h derivative = Sum of roots of 8 8 t h derivative Total number of roots in 8 8 t h derivative \implies \dfrac{\text{Sum of roots of} \ 57^{th} \ \text{derivative}}{\text{Total number of roots in} \ 57^{th} \ \text{derivative}} = \dfrac{\text{Sum of roots of} \ 88^{th} \ \text{derivative}}{\text{Total number of roots in} \ 88^{th} \ \text{derivative}}

⟹ 111 94 − 57 = σ 94 − 88 where σ denotes sum of roots of 8 8 t h derivative \implies \dfrac{111}{94-57} = \dfrac{\sigma}{94-88} ~~~~~~~~~~~~~~~~~~~~~~~~~~~ \small \color{#3D99F6}{\text{where} \ \sigma \ \text{denotes sum of roots of} \ 88^{th} \ \text{derivative}}

⟹ 111 37 = σ 6 \implies \dfrac{111}{37} = \dfrac{\sigma}{6}

⟹ σ = 18 \implies \sigma = \boxed{18}

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