Let `P(x)` be a polynomial with real coefficients, Let `a , b in R ,a < b,` be two consecutive roots of `P(x)`. Show that there exists `c` such that `a lt= c lt= b` and `P^(prime)(c)+100P(c)=0.`

Define a polynomial with real coefficients.

Let P(x) be a polynomial with real coefficients such that P(sin^2x) = P(cos^2x) , for all xϵ[0,π/2] . Consider the following statements: I. P(x) is an even function . II. P(x) can be expressed as a polynomial in (2x−1) 2 III. P(x) is a polynomial of even degree . Then

Let f(x) is a cubic polynomial with real coefficients, x in R such that f''(3)=0 , f'(5)=0 . If f(3)=1 and f(5)=-3 , then f(1) is equal to-

Let P(x) be quadratic polynomical with real coefficient such tht for all real x the relation 2(1 + P(x)) = P(x - 1) + P(x + 1) holds. If P(0) = 8 and P(2) = 32 then Sum of all coefficients of P(x) can not be

Let P(x) be quadratic polynomical with real coefficient such tht for all real x(1) the relation 2(1 + P(x)) = P(x - 1) + P(x + 1) holds. If P(0) = 8 and P(2) = 32 then If the range of P(x) is [m, oo) then 'm' is less then

Let p (x) be a polynomial with real coefficient and p (x)-p'(x) =x^(2)+2x+1. Find P (-1).

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) = 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)=1 and P(2)=-5 , then the minimum-number of distinct real zeroes of P(x) is

Let p(x) be a polynomial of degree 6 with leading coefficient unity and p(-x)=p(x)AAxepsilonR . Also (p(1)+3)^(2)+p^(2)(2)+(p(3)-5)^(2)=0 then sqrt(-4-p(0)) is….