Question

Write an equation of a 4th degree polynomial with zeros at 1, -1, 3i, and -3i

Answer 1

f(x) =
`x^(4)` + 8x² - 9

Explanation:

Given the zeros of a polynomial say x = a and x = b, then

the corresponding factors are (x - a) and (x - b)

The polynomial is the the product of the factors

f(x) = (x - a)(x - b)

Given zeros x = 1, x = - 1, x = 3i and x = - 3i, then corresponding factors are

(x - 1), (x + 1), (x - 3i) and (x + 3i), then

f(x) = (x - 1)(x + 1)(x - 3i)(x + 3i) ← expand in pairs using FOIL

= (x² - 1)(x² - 9i²) → note i² = - 1, thus

= (x² - 1)(x² + 9) ← distribute

=
`x^(4)` + 9x² - x² - 9 ← collect like terms

=
`x^(4)` + 8x² - 9