What are all the roots of the function given that one factor is x²?

To find the roots of the function represented by the polynomial f(x) = 4x³ + 4x² + 16x + 16, we can factor out the known factor, which is x².

Using polynomial long division or synthetic division, we divide f(x) by x²:

          4x + 16
        ________________
x² | 4x³ + 4x² + 16x + 16
       - (4x³)
        ________________
              4x² + 16x
             - (4x²)
        ________________
                    16x + 16
                  - (16x)
        ________________
                          16

So, we can express the polynomial as:

f(x) = x²(4x + 16) + 16

To find the roots, we set the function equal to zero:

x²(4x + 16) + 16 = 0

However, since we know that x² is a factor, we can focus on solving:

4x + 16 = 0

Solve for x:

4x = -16

x = -4

The roots of the original polynomial are therefore:

• x = 0 (with multiplicity 2)
• x = -4

In conclusion, the roots of the function given one factor as x² are x = 0 (two times) and x = -4.