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

Let P(x) =x^(2)+ 1/2x + b and Q(x) = x^(2) + cx + d be two polynomials with real coefficients such that P(x) Q(x) = Q(P(x)) for all real x. Find all the real roots of P(Q(x)) =0.

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 p(x) = a_0+a_1x+ a_2x^2+.............+a_n x^n be a non zero polynomial with integer coefficient . if p( sqrt(2) + sqrt(3) + sqrt(6) )=0 , the smallest possible value of n . is

Let P(x)=x^4+a x^3+b x^2+c x+d be a polynomial such that P(1)=1,P(2)=8,P(3)=27 ,P(4)=64 then the value of 152-P(5) is____________.

Given P(x) =x^(4) +ax^(3) +bx^(2) +cx +d such that x=0 is the only real root of P'(x) =0 . If P(-1) lt P(1), then in the interval [-1,1]

If P (x) is polynomial of degree 4 such than P (-1)=P (1) =5 and P (-2) =P(0)=P (2) =2 find the maximum vaue of P (x).

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

The minimum number of real roots of equation (p''(x))^(2)+p'(x).p'''(x)=0

p(x) be a polynomial of degree at most 5 which leaves remainder - 1 and 1 upon division by (x-1)^3 and (x+1)^3 respectively, the number of real roots of P(x) = 0 is