Find a polynomial equation with real coefficients that has the given roots.
3 and -8i
write the polynomial equation with roots 3 and -8i
Since -8i is a root of the equation, its conjugate, 8i, must also be a root. So the polynomial equation with roots 3, -8i, and 8i is:
(x - 3)(x + 8i)(x - 8i)
Simplifying the expression, we get:
(x - 3)(x^2 + 64)
Expanding the product:
x^3 - 3x^2 + 64x - 3x^2 + 9x - 192
Combining like terms, the polynomial equation with the given roots is:
x^3 - 6x^2 + 73x - 192