(b) Let f be a Cn function on an open interval [a, b]. Assume that the set of zeros {x € (a, b) : f(x) = 0} has at least (n + 1) elements. Prove that there exists xo E (a, b), such that f(m) (xo) = 0. Use (b) to prove that a polynomial (with real coefficients) of degree n has at most n real roots.

Let f(x) = 4x4+x3+2x+1 and g(x) = 3x2+2x+3 be two polynomials in Z5[x].  Determine the quotient and remainder upon dividing f(x) by g(x) in Z5[x].