Write a polynomial equation with roots 5 and -9i
Since complex roots always come in conjugate pairs, the other root will be 9i.
To find the polynomial equation, we need to make use of the fact that if a number x is a root of a polynomial equation, then (x - root) is a factor of the polynomial.
Therefore, the polynomial equation with roots 5, -9i, and 9i can be written as:
(x - 5)(x - (-9i))(x - 9i) = 0
Simplifying this equation:
(x - 5)(x + 9i)(x - 9i) = 0
Expanding the expression using the difference of squares:
(x - 5)(x^2 - (9i)^2) = 0
(x - 5)(x^2 - (-81)) = 0
(x - 5)(x^2 + 81) = 0
Therefore, the polynomial equation with roots 5 and -9i is:
(x - 5)(x^2 + 81) = 0