Algebra 2:)

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f(x)=x^5-2x^4-2x^3+4x^2-3x+3

Approximate the irrational solutions correct to 3 decimals. If there is more than 1 real solution, enter them from smallest to largest, separated by a comma.

  • Algebra 2:) -

    a little synthetic division shows that all the real roots are between -2 and 3.

    Descartes' Rule of Signs shows that there are at most
    3 positive roots
    1 negative root

    Since complex roots appear in pairs, if there's one negative root, there can only be 2 positive roots.

    So, try bisection
    f(-2) = -23
    f(-1) = 9
    f(-1.5) = 5.5
    f(-1.75) = -3.9
    f(-1.625) = 1.743
    f(-1.6875) = -0.839
    f(-1.65625) = 0.515
    f(-1.671875) = -0.146
    and so on till you get f(-1.6685) = 0

    proceed likewise for the other roots.

    -1.668, 1.208, 2.244

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