Answer:
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
