solve the polynomial equation: x^3+64
do no round. leave answers in exact form.
sort them in this format: x= blank and blank
To solve the polynomial equation x^3 + 64 = 0, we can use the fact that the sum of two cubes formula is given by:
a^3 + b^3 = (a + b)(a^2 - ab + b^2)
In this case, we have x^3 + 4^3 = (x + 4)(x^2 - 4x + 16)
Setting each factor equal to zero:
x + 4 = 0
x = -4
x^2 - 4x + 16 = 0
Applying 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
As we have a negative under the square root, the solutions will involve complex numbers. Continuing the calculation:
x = (4 ± √48i) / 2
x = (4 ± (√16)(√3)i) / 2
x = (4 ± 4√3i) / 2
x = 2 ± 2√3i
Therefore, the solutions to the polynomial equation x^3 + 64 = 0 are x = -4 and x = 2 ± 2√3i.