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
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1 answer

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
57i and 5+7i
sum = 10
product = (57u)(5+7i) = 25  49i^2 = 25+49 = 74
equation must be
x^2  10x + 74 = 0
or .... if the roots are 57i and 5+7i
then (x  (57i))(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 👍
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