write a polynomial function to the nth degree that has the solutions

Question

the solutions are, n=3 x=2 x=4i

help me best to show work/steps to solve

Answer 1

Solutions: x = 3, x = 2, x = 4i

Factors: x - 3 = 0; x - 2 = 0; x - 4i = 0; x + 4i = 0 (This needs to have a conjugate pair)

(x - 3)(x - 2) = x² - 2x - 3x + 6 = x² - 5x + 6 [Distribute both x and -3 to (x - 2) such as x(x - 2) - 3(x - 2)]

(x - 4i)(x + 4i) = x² + 4ix - 4ix - 16i² = x² - 16(-1) = x² + 16

[Distribute both x and -4i to (x + 4i) such as x(x + 4i) - 4i(x + 4i)]

(x² + 16)(x² - 5x + 6) = x²(x² - 5x + 6) + 16(x² - 5x + 6)

= x⁴ - 5x³ + 6x² + 16x² - 80x + 96

= x⁴ - 5x³ + 22x² - 80x + 96

Answer 2

This is a 3rd degree polynomial so it will have no more than 3 solutions. Imaginary solutions always occur in pairs so your third solution is -4i so your equation in factored form is y= (x-2)(x+4i)(x-4i)

(x+4i)(x-4i) = (x^2 +16)

(x^2+16)(x-2) = x^3 -2x^2 +16x -32