Question

Solve the equation

x3 - 4x2 + 16x - 64 = 0 by finding all the roots if 4
is one of the roots.

Answer 1

-4iand4i

Explanation:

Answer 2

3 roots are:

4, 4i, -4i

Explanation:

This is a cubic equation that has 3 roots. One root is given, we got to find the other two.

Let's group first 2 terms and last two terms and factor and solve:

`x^3 - 4x^2 + 16x - 64 = 0\\x^2(x-4) + 16(x-4)=0\\(x^2+16)(x-4)=0`

From here we can say:

x^2 + 16 = 0

and

x - 4 = 0 [x = 4, we already know this solution]

Let's find the other 2 roots from the 1st equation:

`x^2 + 16 = 0\\x^2=-16\\x=+-4i`

Note:
`√(-1)=i`

So the 3 roots are:

4, 4i, -4i