# Roots

1. Find the number of real roots and imaginary roots:

f(x)=10x^5-34x^4-5x^3-8x^2+3x+8

2. Find all zeros:

f(x)=x^4+2x^3+5x^2+34x+30

3. Find all roots:

f(x)=x^3-7x^2-17x-15; 2 + i

4. Find all roots:

f(x)=x^4-6x^3+12x^2+6x-13; 3 + 2i

The 5th problem is attached.

Please answer the questions above and show the work done. Thank you!

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#### Solution Preview

See attachment

1.Find the number of real roots and imaginary roots:

In order to find all real roots, we must count the number of times the sign changes in the equation for f(x). Thus, there are two sign changes, from +10 to -34, and -8 to +3. Thus, there are two positive real roots.

In order to find the number of negative real roots, we need to count the sign changes in f(-x). Thus,

Thus, the sign changes from -34 to +5, +5 to -8, and ...

#### Solution Summary

This shows how to find real and imaginary roots for polynomials up to degree 5.