Question

What are the real and complex solutions of the polynomial equation X^3-216=0

Answer 1

Answer:

$6,\ -3+3\sqrt{3}i,\ -3-3\sqrt{3}i$

Step-by-step explanation:

Factor the equation $x^3-216=0$ using formula for difference of the cubes:

$x^3-216=(x-6)(x^2+6x+36),$

then

$x-6=0\ or\ x^2+6x+36=0.$

1. The equation $x-6=0$ has real solution $x=6.$

2. The equation $x^2+6x+36=0$ has negative discriminant $D=6^2-4\cdot 36=36-144=-108,$ then it has two complex solutions

$x_{1,2}=\dfrac{-6\pm 6\sqrt{3}i}{2}=-3\pm3\sqrt{3}i.$