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Repeated roots : If equation f(x) = 0, where f(x) is a polyno- mial function, has roots alpha,alpha,beta,… or alpha root is repreated root, then f(x) = 0 is equivalent to (x-alpha)^(2)(x-beta)…=0, from which we can conclude that f(x)=0 or 2(x-alpha)[(x-beta)...]+(x-alpha)^(2)[(x-beta)...]'=0 or (x-alpha) [2 {(x-beta)...}+(x-alpha){(x-beta)...}']=0 has root alpha. Thus, if alpha root occurs twice in the, equation, then it is common in equations f(x) = 0 and f'(x) = 0. Similarly, if alpha root occurs thrice in equation, then it is common in the equations f(x)=0, f'(x)=0, and f'''(x)=0. If x-c is a factor of order m of the polynomial f(x) of degree n (1ltmltn), then x=c is a root of the polynomial [where f^(r)(x) represent rth derivative of f(x) w.r.t. x] |
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Answer» `F^(m)(x)` g(x) is polynomial of degree n-m. Then x=c is common root for the equations `f(x)=0,f^(1)(x)=0, f^(2)(x)=0,…,f^(m-1)(x)=0` where f'(x) represent rth DERIVATIVE of f(x) w.r.t. x,. |
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