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 + 3i

y = x⁴ - 9x² - 50x - 150

Step-by-step explanation:

Complex roots come in conjugate pairs. So if - 1 + 3i is a root, then - 1 - 3i is also a root.

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

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

Distribute using FOIL (first, outer, inner, last) to get real coefficients:

y = (x - 5) (x + 3) (x² + (1 + 3i) x + (1 - 3i) x + (1 - 3i) (1 + 3i))

y = (x - 5) (x + 3) (x² + x + 3ix + x - 3ix + 1 + 3i - 3i - 9i²)

y = (x - 5) (x + 3) (x² + 2x + 1 + 9)

y = (x - 5) (x + 3) (x² + 2x + 10)

Distribute to convert from factored form to standard form:

y = (x² + 3x - 5x - 15) (x² + 2x + 10)

y = (x² - 2x - 15) (x² + 2x + 10)

y = x² (x² + 2x + 10) - 2x (x² + 2x + 10) - 15 (x² + 2x + 10)

y = x⁴ + 2x³ + 10x² - 2x³ - 4x² - 20x - 15x² - 30x - 150

y = x⁴ - 9x² - 50x - 150