Repeated roots : If equation f(x) = 0, where f(x) is a polyno- mial function, has roots $α, α, β, \dots or α$ root is repreated root, then f(x) = 0 is equivalent to $(x - α)^{2} (x - β) \dots = 0,$ from which we can conclude that $f (x) = 0 or 2 (x - α) [(x - β) \dots] + (x - α)^{2} [(x - β) \dots]^{'} = 0 or (x - α) [2 {(x - β) \dots} + (x - α) {(x - β) \dots}^{'}] = 0$ has root $α$ . Thus, if $α$ root occurs twice in the, equation, then it is common in equations f(x) = 0 and f'(x) = 0. Similarly, if $α$ 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 $(1 < m < n)$ , then x=c is a root of the polynomial [where $f^{r} (x)$ represent rth derivative of f(x) w.r.t. x]

Question

Repeated roots : If equation f(x) = 0, where f(x) is a polyno- mial function, has roots

α, α, β, \dots or α

root is repreated root, then f(x) = 0 is equivalent to

(x - α)^{2} (x - β) \dots = 0,

from which we can conclude that

f (x) = 0 or 2 (x - α) [(x - β) \dots] + (x - α)^{2} [(x - β) \dots]^{'} = 0 or (x - α) [2 {(x - β) \dots} + (x - α) {(x - β) \dots}^{'}] = 0

has root

α

. Thus, if

α

root occurs twice in the, equation, then it is common in equations f(x) = 0 and f'(x) = 0. Similarly, if

α

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

(1 < m < n)

, 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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