solve the polynomial equation: x^3+64
do no round. leave answers in exact form.
x= blank and blank
We have the polynomial equation x^3 + 64 = 0.
To solve this equation, we can rewrite it as (x + 4)(x^2 - 4x + 16) = 0.
Setting each factor to zero gives us two possible solutions:
1) x + 4 = 0 => x = -4
2) x^2 - 4x + 16 = 0
Using the quadratic formula, we find the roots of the second factor:
x = (-(-4) ± √((-4)^2 - 4*1*16)) / (2*1)
x = (4 ± √(16 - 64)) / 2
x = (4 ± √(-48)) / 2
x = (4 ± √(48) * √(-1)) / 2
x = (4 ± 4√3i) / 2
x = 2 ± 2√3i
Therefore, the solutions to the given polynomial equation are:
x = -4, 2 + 2√3i, 2 - 2√3i.