Let `P(x)` be a polynomial with real coefficients, Let `a , b in R ,a

Let P(x)=x^3-8x^2+c x-d be a polynomial with real coefficients and with all it roots being distinct positive integers. Then number of possible value of c is___________.

Let f(x) be a polynomial with integral coefficients. If f(1) and f(2) both are odd integers, prove that f(x) = 0 can' t have any integral root.

Let y=f(x) be a polynomial of odd degree (geq3) with real coefficients and (a, b) be any point. Statement 1: There always exists a line passing through (a , b) and touching the curve y=f(x) at some point. Statement 2: A polynomial of odd degree with real coefficients has at least one real root.

Let f(x) = x^4 + ax^3 + bx^2 + cx + d be a polynomial with real coefficients and real roots. If |f(i)|=1where i=sqrt(-1) , then the value of a +b+c+d is

Let P(x) = x^10+ a_2x^8 + a_3 x^6+ a_4 x^4 + a_5x^2 be a polynomial with real coefficients. If P(1) and P(2)=5 , then the minimum-number of distinct real zeroes of P(x) is

A nonzero polynomial with real coefficient has the property that f(x)=f^(prime)(x)dot f^(prime prime)(x)dot If a is the leading coefficient of f(x), then the value of 1/(2a) is____

Let f(x), g(x) , and h(x) be the quadratic polynomials having positive leading coefficients and real and distinct roots. If each pair of them has a common root, then find the roots of f(x) + g(x) + h(x) = 0 .

Let p(x)=0 be a polynomial equation of the least possible degree, with rational coefficients having ""^(3)sqrt7 +""^(3)sqrt49 as one of its roots. Then product of all the roots of p(x)=0 is a. 56 b. 63 c. 7 d. 49

Let p(x) be a real polynomial of least degree which has a local maximum at x=1 and a local minimum at x=3. If p(1)=6a n dp(3)=2, then p^(prime)(0) is_____

Let f(x) be a polynomial function of second degree. If f(1) = f(-1) and a, b. c are in A.P., then f'(a), f'(b),f'(c) are in