Consider the quadratic polynomial qx x2 ax b find a

Question

1: Determine all polynomials x² + px + q with roots x = p and x = q.

2: (a) Consider the quadratic polynomial, Q(x) = x² + ax + b. Find a polynomial whose roots are the squares of the roots of Q(x).

(b) Consider the cubic polynomial, C(x) = x³ + ax² + bx + c. Find a polynomial whose roots are the cubes of the roots of C(x).

3: If P_n(x) denotes a polynomial of degree n such that P_n(k) = 1/k for k = 1, 2, 3, . . . , n+1, determine P_n(n + 2).

4: (a) For any real number c and any j ∈ {1, 2, 3, 4}, let r_c,j be the number of real roots of the polynomial x⁵ + cx^j. What is the maximum possible value of r_c,j over all real values of c and all j ∈ {1, 2, 3, 4}?

(b) Let k be the smallest positive integer for which there exist distinct integers m₁, m₂, m₃, m₄, m₅ such that the polynomial p(x) = (x - m₁) (x - m₂) (x - m₃) (x - m₄) (x - m₅), has exactly k nonzero coefficients. Find, with proof, a set of integers m₁, m₂, m₃, m₄, m₅ for which this minimum k is achieved.

5: A student walks into a classroom and finds a partially erased problem on the blackboard which states: "The following polynomial of degree twenty has all real positive roots. Find them." However the only terms left are:

x²⁰ - 20x¹⁹ + 1

Can you solve the problem anyway?