Question

For which values of a and b , are the zeroes of q(x)=x3+ 2x2+a also the zeroes of the polynomial p (x)= x5 -x4-4x3+3x2+3x+b? Which zeroes of p (x) are not the zeroes of q (x)?

Answer 1

If the roots of q are roots of p, then q must divide p. If you do the long division, you will find that the remainder is

-(a+1)x^2 + 3(a+1)x + (b-2a)

These must all be zero, which means that
a = -1
b = -2

So,
p(x) = x^5-x^4-4x^3+3x^2+3x-2
q(x) = x^3+2x^2-1

q(x) = (x+1)(x^2+x-1)
p(x) = (x+1)(x^2+x-1)(x-1)(x-2)