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]

Options

A. If `pltqltn`, then `alpha" and "beta` are two of the roots of the equation `f^(p-1)(x)=0.`
B. If `qltpltn`, then `alpha" and "beta` are two of the roots of the equation `f^(q-1)(x)=0.`
C. If `pltqltn,` then equations f(x) = 0 and `f^(p)(x)=0` have exactly one root common
D. If `qltpltn,` then equations `f^(q)(x)=0" and "f^(p)(x)=0` have exactly two roots common.