What are the real and complex solutions of the polynomial equation x³ - 64 = 0
4, -1 + 2i Sqrt3, -1 + 2i sqrt3
4, 1 +2i sqrt3 , 1 + 2i sqrt3
4, -2 + 2i sqrt3 -2 - 2i sqrt3
4, -2 + 2i sqrt3, 2 + 2i sqrt3

Solution:

Given polynomial equation is x³ - 64 = 0

x³ - 64 is the cubic polynomial in the form a³ - b³ = (a - b)(a² + ab + b²)

x³ - 4³ = (x - 4)(x² + 4x + 4²)

One root is 4

x² + 4x + 16 = 0

x = -b ± √(b² - 4ac) / 2a

x = -4 ± √4² - 64 /2

x = -2 ± √-48 /2

x = -2 ± 4√3i /2

x = -2 ± 2√3i

Therefore, x = 4, -2 ± 2√3i

What are the real and complex solutions of the polynomial equation x³ - 64 = 0

Summary:

The real and complex solutions of the polynomial equation x³ - 64 = 0 are x = 4, -2 ± 2√3i