math (precalc)

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Find a polynomial with integer coefficients that satisfies the given conditions.
R has degree 4 and zeros 3 − 4i and 5, with 5 a zero of multiplicity 2.

  • math (precalc) -

    x = 3 - 4i
    x = 3 + 4i
    x = 5
    x =5

    two 5's because multiplicity 2
    You always have to have the conjugate of an imaginary number.

    (x -(3-4i))(x-(3+4i)(x-5)(x-5) = 0
    Easiest way is to multiply the first two and the second two factors.

    Then multiply those answers together to get the polynomial.

  • math (precalc) -

    (x-5)(x-5)(x-3+4i)(x-3-4i) = 0

    (x^2-10x+25)(x^2-6x+25)

    multiply that out

  • math (precalc) -

    complex values come in pairs (why?), so the roots are

    3-4i, 3+4i, 5, 5

    R(x) = (x-(3-4i))(x-(3+4i))(x-5)^2
    = ((x-3)+4i)((x-3)-4i)(x-5)^2
    = ((x-3)^2+4^2)(x-5)^2
    = (x^2-6x+25)(x-5)^2
    = x^4 - 16x^3 + 110x^2 - 400x + 625

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