If `x-r` is a factor of the polynomial `f(x)=a_0x^n+a_1x^(n-1)+.....+a_n` repeated `m` times, `(1ltxlt=n),` then `r` is a root of `f^(prime)(x)=0`repeated `m` times.

If x-c is a factor of order m of the polynomial f(x) of degree n (1 < m < n) , then find the polynomials for which x=c is a root.

Find the coefficient of x^n in the polynomial (x+^n C_0)(x+3^n C_1)xx(x+5^n C_2)[x+(2n+1)^n C_n]dot

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 alpha root occurs p times and beta root occurs q times in polynomial equation f(x)=0 of degree n(1ltp,qltn) , then which of the following is not ture [where f^(r)(x) represents rth derivative of f(x) w.r.t x] ?

Let f(x0 be a non-constant thrice differentiable function defined on (-oo,oo) such that f(x)=f(6-x)a n df^(prime)(0)=0=f^(prime)(x)^2=f(5)dot If n is the minimum number of roots of (f^(prime)(x)^2+f^(prime)(x)f^(x)=0 in the interval [0,6], then the value of n/2 is___

If f(x)={0 ,where x=n/(n+1),n=1,2,3.... and 1 else where then the value of int_0^2 f(x) dx is

If f(x)=|x a a a x a a a x|=0, then f^(prime)(x)=0a n df^(x)=0 has common root f^(x)=0a n df^(prime)(x)=0 has common root sum of roots of f(x)=0 is -3a none of these

If f(x) =(p-x^n)^(1/n) , p >0 and n is a positive integer then f[f(x)] is equal to

f(x) is cubic polynomial with f(x)=18a n df(1)=-1 . Also f(x) has local maxima at x=-1a n df^(prime)(x) has local minima at x=0 , then the distance between (-1,2)a n d(af(a)), where x=a is the point of local minima is 2sqrt(5) f(x) is increasing for x in [1,2sqrt(5]) f(x) has local minima at x=1 the value of f(0)=15

If f(x)=x/(sinx)a n dg(x)=x/(tanx),w h e r e0ltxlt=1, then in this interval