Divide the polynomial p(x) = x⁴ – 3x² + 4x + 5 by g(x) = x² + 1 – x and find the quotient and remainder.

If and x = –3 are the roots of the quadratic equation ax² + 7x + b = 0 then find the values of a and b.

Find the zeroes of g(x) = a(x² + 1) – x(a² + 1) and verify the relationship between zeroes of polynomial and its coefficient.

Check whether the first polynomial is a factor of the second polynomial by dividing the second polynomial by the first polynomial:

P(x) = x³ – 3x + 1 and q(x) = x⁵ – 4x³ + x² + 3x + 1

On dividing x³ – 3x² + x + 2 by a polynomial g(x), the quotient and remainder were (x – 2) and (–2x + 4) respectively. Find g(x).