Question

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

Answer 1

y = x⁴ − 9x² − 50x − 150

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