Repeated roots: If equation

MATHEMATICS · Single correct · CBSE · Mathematics · Continuity and Differentiability · Continuity and Differentiability

Problem

Repeated roots: If equation f(x)=0f(x)=0, where f(x)f(x) is a polynomial function, has roots α,α,β,…\alpha,\alpha,\beta,\dots or α\alpha root is repeated root, then f(x)=0f(x)=0 is equivalent to (x−α)2(x−β)⋯=0(x-\alpha)^2(x-\beta)\dots=0, from which we can conclude that f′(α)=0f'(\alpha)=0 or 2(x−α)[(x−β)… ]+(x−α)2[(x−β)… ]′=02(x-\alpha)[(x-\beta)\dots]+(x-\alpha)^2[(x-\beta)\dots]'=0 or (x−α)[2{(x−β)… }+(x−α){(x−β)… }′]=0(x-\alpha)[2\{(x-\beta)\dots\}+(x-\alpha)\{(x-\beta)\dots\}']=0 has root α\alpha. Thus, if a root occurs twice in the equation, then it is common in equations f(x)=0f(x)=0 and f′(x)=0f'(x)=0. Similarly, if a root occurs thrice in equation, then it is common in the equations f(x)=0f(x)=0, f′(x)=0f'(x)=0, and f′′(x)=0f''(x)=0. If x−cx-c is a factor of order mm of the polynomial f(x)f(x) of degree nn (1<m<n1<m<n), then x=cx=c is a root of the polynomial. [where fr(x)f^r(x) represent rrth derivative of f(x)f(x) w.r.t. xx ] (a) fm(x)f^m(x) (b) fm−1(x)f^{m-1}(x) (c) fn(x)f^n(x) (d) none of these

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