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

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

Let f(x) =4x^2-4ax+a^2-2a+2 be a quadratic polynomial in x,a be any real number. If x-coordinate of vertex of parabola y =f(x) is less than 0 and f(x) has minimum value 3 for x in [0,2] then value of a is

Suppose p(x)=a_0+a_1x+a_2x^2++a_n x^ndot If |p(x)|lt=e^(x-1)-1| for all xgeq0, prove that |a_1+2a_2++n a_n|lt=1.

Let f(x)=a_5x^5+a_4x^4+a_3x^3+a_2x^2+a_1x , where a_i ' s are real and f(x)=0 has a positive root alpha_0dot Then (a) f^(prime)(x)=0 has a positive root alpha_1 such that 0alpha_1alpha_0

Let f(x)=a_5x^5+a_4x^4+a_3x^3+a_2x^2+a_1x , where a_i ' s are real and f(x)=0 has a positive root alpha_0dot Then f^(prime)(x)=0 has a positive root alpha_1 such that 0

If (1+x^(2))^(2)(1+x)^(n) =a_(0) + a_(1)x + a_(2)x^(2) + …+ x^(n+4) and if a_(0), a_(1), a_(2) are in A.P., then n is:

If (1+x+x^2++x^p)^n=a_0+a_1x+a_2x^2++a_(n p)x^(n p), then find the value of a_1+2a_2+3a_3+ddot+n pa_(n p)dot

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 C_1 and C_2 be two circles whose equations are x^2+y^2-2x=0 and x^2+y^2+2x=0 and P(lambda, lambda) is a variable point