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 a_(1)x^(3)+b_(1)x^(2)+c_(1)x+d_(1)=0 and a_(2)x^(3)+b_(2)x^(2)+c_(2)x+d_(2)=0 have a pair of repeated roots common, then |{:(3a_(1),2b_(1),c_(1)),(3a_(2),2b_(2),c_(2)),(a_(2)b_(1)-a_(1)b_(2),c_(1)a_(2)-c_(2)a_(1),d_(1)a_(2)-d_(2)a_(1)):}|=
