Repeated roots: If equation f(x)=0, where f(x) is a polynomial function, has roots α,α,β,… or α root is repeated root, then f(x)=0 is equivalent to (x−α)2(x−β)⋯=0, from which we can conclude that f′(α)=0 or 2(x−α)[(x−β)…]+(x−α)2[(x−β)…]′=0 or (x−α)[2{(x−β)…}+(x−α){(x−β)…}′]=0 has root α. Thus, if a root occurs twice in the equation, then it is common in equations f(x)=0 and f′(x)=0. Similarly, if a root occurs thrice in equation, then it is common in the equations f(x)=0, f′(x)=0, and f′′(x)=0.
If α root occurs p times and β root occurs q times in polynomial equation f(x)=0 of degree n (1<p,q<n), then which of the following is not true [where fr(x) represents rth derivative of f(x) w.r.t. x ]?
(a) If p<q<n, then α and β are two of the roots of the equation fp−1(x)=0
(b) If q<p<n, then α and β are two of the roots of the equation fq−1(x)=0.
(c) If p<q<n, then equations f(x)=0 and fp(x)=0 have atleast one root common.
(d) If q<p<n, then equation fq(x)=0 and fp(x)=0 have exactly two roots common.