Write a polynomial function of minimum degree with real coefficients whose zeros include those listed. Write the polynomial in standard form.

5, - 3, and - 1 + 2i

x^4 - 14x^2 - 40x - 75.

Step-by-step explanation:

As complex roots exist in conjugate pairs the other zero is - 1 - 2i.

So in factor form we have the polynomial function:

(x - 5) (x + 3) (x - (-1 + 2i)) (x - (-1 - 2i)

= (x - 5) (x + 3) (x + 1 - 2i) (x + 1 + 2i)

The first 2 factors = x^2 - 2x - 15 and

(x + 1 - 2i) (x + 1 + 2i) = x^2 + x + 2ix + x + 1 + 2i - 2ix - 2i - 4 i^2

= x^2 + 2x + 1 + 4

= x^2 + 2x + 5.

So in standard form we have:

(x^2 - 2x - 15) (x^2 + 2x + 5)

= x^4 + 2x^3 + 5x^2 - 2x^3 - 4x^2 - 10x - 15x^2 - 30x - 75

= x^4 - 14x^2 - 40x - 75.