Question

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

Answer 1

`x^(4)` + 3x² - 4

Explanation:

Note that complex zeros occur in conjugate pairs

If 2i is a zero then - 2i is a zero

The zeros are x = 1, x = - 1, x = 2i, x = - 2i, thus the factors are

(x - 1), (x + 1), (x - 2i) and (x + 2i)

The polynomial is expressed as the product of the factors, thus

f(x) = (x - 1)(x + 1)(x - 2i)(x + 2i) ← expanding in pairs

= (x² - 1)(x² - 4i²) → i² = - 1

= (x² - 1)(x² + 4) ← distribute

=
`x^(4)` + 4x² - x² - 4

=
`x^(4)` + 3x² - 4