What are the real and complex solutions of the polynomial equation?
×^3 - 64 =0
(1 point)
O 4, -1+25,-1+2iN3
O 4,1+213,1+213
O 4,-2+25,-2-2iN5
The polynomial equation x^3 - 64 = 0 can be factored as (x - 4)(x^2 + 4x + 16) = 0.
Setting each factor equal to zero, we get:
x - 4 = 0 --> x = 4 (real solution)
x^2 + 4x + 16 = 0
Using the quadratic formula, we find the complex solutions for x^2 + 4x + 16 = 0:
x = (-4 ± √(4^2 - 4(1)(16))) / (2(1))
x = (-4 ± √(-48)) / 2
x = (-4 ± 4√3i) / 2
x = -2 ± 2√3i
So, the real solution is 4, and the complex solutions are -2 + 2√3i and -2 - 2√3i.
Answer: O 4,-2+2√3i,-2-2√3iN3