Write a polynomial function in standard form given the zeros.

-1/2, - 5 + i

y = x³ + 10.5x² + 31x + 13

Step-by-step explanation:

Complex roots (roots that have imaginary terms) always come in conjugate pairs. So if one root is - 5 + i, there's another root that's - 5 - i.

So the polynomial is:

y = (x + 1/2) (x - (-5 + i)) (x - (-5 - i))

Distributing:

y = (x + 1/2) (x² - (-5 + i) x - (-5 - i) x + (-5 + i) (-5 - i))

y = (x + 1/2) (x² + 5x - ix + 5x + ix + (-5 + i) (-5 - i))

y = (x + 1/2) (x² + 10x + (-5 + i) (-5 - i))

y = (x + 1/2) (x² + 10x + 25 + 5i - 5i - i²)

y = (x + 1/2) (x² + 10x + 25 + 1)

y = (x + 1/2) (x² + 10x + 26)

y = x (x² + 10x + 26) + 1/2 (x² + 10x + 26)

y = x³ + 10x² + 26x + 1/2x² + 5x + 13

y = x³ + 10.5x² + 31x + 13