Identify the simplest polynomial function having integer coefficients with the given zeros: 3i, −1, 2

MUST SHOW WORK

9514 1404 393

  f(x) = x^4 -x^3 +7x^2 -9x -18

Step-by-step explanation:

The imaginary root has a corresponding conjugate that is also a root. Each root has a corresponding factor: (x -p) for root p.

  f(x) = (x -3i)(x +3i)(x -(-1))(x -2)

  f(x) = (x^2 +9)(x^2 -x -2) . . . . combine pairs of factors

  f(x) = x^4 -x^3 +7x^2 -9x -18 . . . . finish multiplying it out

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We have made use of the products of binomials:

  (x +a)(x +b) = x^2 +(a+b)x +ab

  (x +a)(x -a) = x^2 -a^2

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