Question

What is the polynomial function of lowest degree with lead coefficent 1 and roots i, -2, 2?

Answer 1

`x^(4)` - 3x² - 4

Explanation:

given the roots of a polynomial, say x = a, x = b and x = c

Then the factors of the polynomial are (x - a), (x - b) and (x - c)

and f(x) = a(x - a)(x - b)(x - c) ← a is a multiplier

Note that complex roots occur in conjugate pairs

hence x = i is a root then x = - i is also a root

the roots are x = - 2, x = 2, x = i and x = - i

factors are (x + 2)(x - 2)(x - i)(x + i) ← expand in pairs

f(x) = (x² - 4)(x² - i²) → (i² = - 1 )

= (x² - 4(x² + 1)

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

=
`x^(4)` - 3x² - 4