how to write a polynomial function of least degree with integral coefficients that has the given zeros.

Question

The zeros are 3i, 2i, and 1.

Answer 1

Solutions: x = 3i, x = 2i, x = 1, x = -3i, x = -2i (This comes with conjugate pairs)

Factors: x - 3i = 0; x + 3i = 0; x - 2i = 0; x + 2i = 0; x - 1 = 0

(x - 3i)(x + 3i) = x² + 3ix - 3ix - 9i² = x² - 9(-1) = x² + 9

(x - 2i)(x + 2i) = x² + 2ix - 2ix - 4i² = x² - 4(-1) = x² + 4

(x - 1)(x² + 4)(x² + 9) = (x - 1)(x⁴ + 9x² + 4x² + 36)

= (x - 1)(x⁴ + 13x² + 36)

= x(x⁴ + 13x² + 36) - 1(x⁴ + 13x² + 36)

= x⁵ + 13x³ + 36x - x⁴ - 13x² - 36

= x⁵ - x⁴ + 13x³ - 13x² + 36x - 36