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Determine the different possibilities as to the kind of roots that x9 – 4x8 +7x6+x5-3x4+x2-2x+2 will have. Show complete solution and explain the answer.

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Because of the Fundamental Theorem of Algebra there are 8 roots possibly complex and posiblly not unique.

The rantional roots theorem says that any rational roots would be `+- 2` or `+- 1` .

y=x^9-4x^8+7x^6+x^5-3x^4+x^2-2x+2

If you graph the function you can see there are three real roots around -1.124, 1.692, and 3.386, and the other 6 roots are complex.

Since the coefficients are real and by the complex solutions theorem, the complex solutions are 3 pairs of complex numbers that are complex conjugates of one another.  The three pairs are (to 3 decimal places):

`0.0835 +- 0.703i` ,  `0.728 +- 0.327i` , and `-0.788 +- 0.594`

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