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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)?

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  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)

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