Question

Find a polynomial equation with real coefficients that has the given roots.

5 and -7i

Write a polynomial equation with roots 5 and -7i

x^3 - __ x^2 + __ x -__ = 0

Answer 1

Don't know if they still teach that, but ....
for any quadratic equation of the form
ax^2 + bx + c = 0
the sum of the roots is -b/a , and the product of the roots is c/a

complex roots always appear as conjugate pairs, so they must be
5-7i and 5+7i
sum = 10
product = (5-7u)(5+7i) = 25 - 49i^2 = 25+49 = 74

equation must be
x^2 - 10x + 74 = 0

or .... if the roots are 5-7i and 5+7i
then (x - (5-7i))(x - (5+7i)) = 0
(x - 5 + 7i)(x - 5 - 7i) = 0
x^2 - 5x - 7ix - 5x + 25 + 35i + 7ix - 35i - 49i^2 = 0
x^2 - 10x + 25 - 49i^2 = 0 , but i^2 = -1

x^2 - 10x + 74 = 0