Suppose p is a polynomial with complex coefficients and all even degreen. If all the roots of p are complex non-real numbers with modulus 1, prove that ` p(1) in R ``iff` ` p(-1) in R `

Find all polynomials whose coefficients are equal to 1 or - 1 and whose all roots are real.

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

Prove that P(n,r) = (n- r+1) P(n,r-1)

Let P(x) = x^(2) + ax + b be a quadratic polynomial with real coefficients. Suppose there are real numbers s ne t such that P(s) = t and P(t) = s. Prove that b - st is a root of the equation x^(2) + ax + b st =0 .

If all the roots of x^3 + px + q = 0 p,q in R,q != 0 are real, then

Let R be the relation on the set R of all real numbers defined by a R b Iff |a-b| le1. Then R is

if p: q :: r , prove that p : r = p^(2): q^(2) .

If P(n) is the statement 2^ngeq3n , and if P(r) is true, prove that P(r+1) is true.

Let P be the relation defined on the set of all real number such that P = [(a, b) : sec^2 a-tan^2 b = 1] . Then P is:

Prove that ^(n-1) P_r+r .^(n-1) P_(r-1) = .^nP_r