Question
Write a polynomial equation with roots 5 and -9i
Answers
Answered by
GPT 3.5
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
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