What are the real and complex solutions of the polynomial equation? x^3−64=0
To find the solutions of the polynomial equation x^3 - 64 = 0, we can factor it as a difference of cubes:
(x - 4)(x^2 + 4x + 16) = 0
From the first factor, we get: x - 4 = 0
x = 4
For the second factor, we can use the quadratic formula to solve for x:
x = (-4 ± √(4^2 - 4(1)(16))) / (2*1)
x = (-4 ± √(16 - 64)) / 2
x = (-4 ± √(-48)) / 2
x = (-4 ± 4√(-3)) / 2
x = -2 ± 2√(-3)
Thus, the complex solutions are x = -2 + 2√(-3) and x = -2 - 2√(-3), while the real solution is x = 4.