SOLUTION
We want to factor
[tex]x^3-64=0[/tex]
Looking at this, we can tell that (x - 4) or (x + 4) would be one of its factors, since 4 is a factor of 64. So let use check for (x - 4)
So, we will put x = 4 into the equation, we have
[tex]\begin{gathered} x^3-64 \\ 4^3-64 \\ 64-64=0 \end{gathered}[/tex]
hence (x - 4) is a factor. Dividing the polynomial by (x - 4), we have
[tex]\frac{x^3-64}{x-4}[/tex]
so we got
[tex]x^2+4x+16[/tex]
Factorizing the result, we have
[tex]\begin{gathered} x^2+4x+16 \\ We\text{ find the discriminant using } \\ D=b^2-4ac \\ D=4^2-4\times1\times16 \\ D=16-64 \\ D=-48 \end{gathered}[/tex]
Now we have the discriminant, we use the formula to fin the roots of this equation, we have
[tex]\begin{gathered} x_1=\frac{-b-\sqrt{D}}{2a} \\ =\frac{-4-\sqrt{-48}}{2\times1} \\ =\frac{-4-4\sqrt{3}i}{2} \\ =-2-2\sqrt{3}i \end{gathered}[/tex]
The second root becomes
[tex]\begin{gathered} x_1=\frac{-b+\sqrt{D}}{2a} \\ =\frac{-4+\sqrt{-48}}{2\times1} \\ =\frac{-4+4\sqrt{3}i}{2} \\ =-2+2\sqrt{3}i \end{gathered}[/tex]
Note that square root of -1 is i
So, comparing to the options, we can see that
The answer is option D
