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)=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 of degree 5 such that f(|x|)=0 has 8 real distinct , Then number of real roots of f(x)=0 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 find the remainder when P(5) is divided by 5.

If x= 3p and y=(2p)/5+1 , then find the value of p for which x = 5y.

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

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.Then: 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 .

If p(x) be a polynomial satisfying the identity p(x^2)+2x^2+10x=2xp(x+1)+3 , then p(x) is given by

Let X and Y be two events such that P(X)=1/3, P(X|Y)=1/2and P(Y|X)=2/5. Then

Let P(x)=a_0+a_1x^2+a_2x^4++a_n x^(2n) be a polynomial in a real variable x with 0 < a_0 < a_1 < a_2 <......< a_n . The function P(x) has a. neither a maximum nor a minimum b. only one maximum c. only one minimum d. only one maximum and only one minimum e. none of these

If x be a factor of the polynomial p(x) = (x - 1) (x - 2) (x - 3), then p(0) =