Exercise 4.

If you have a device to compute roots numerically: Find all roots of the polynomial

How many complex roots does the polynomial have?

Answer.

Mathematica lists the following approximations for the roots:

This is correct; the polynomial can be factored as

The term accounts for the triple occurrence of the real root x=2. indeed has the four complex roots listed above.

My graphing calculator, the TI-85, gets it wrong: It comes up with only one real root x=2, the other six roots it lists are complex. Admittedly, two of the complex roots have a very tiny imaginary part (the number in front of the i), so they are "nearly" real.

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