Question

Write a polynomial function of least degree with integral coefficients that has the given zeros.


0, -5, 3 + i

need to show all work

Answer 1

The zeroes are
0, -5, 3 + i (and its conjugate)

The factors are
x (x + 5) (x - (3 + i)) (x - (3 - i))
Expanding...
(x² + 5x)((x - 3) - i) ((x - 3) + i)
(x² + 5x)((x - 3)² - i²)
(x² + 5x)(x² - 6x + 9 - (-1))
(x² + 5x)(x² - 6x + 10)
x⁴ - 6x³ + 10x² + 5x³ - 30x² + 50x
x⁴ - x³ - 20x² + 50x