6. Use Descartes' rule of signs to describe the roots of .

Part I: How many total roots must there be in this fourth-degree function? (2 points)

Part II:

a. How many sign changes are there in the original function H(x)? (1 point)

b. How many positive real roots are possible? (2 points)

Part III:

a. Replace every x with (-x) and rewrite the function below. (2 points)

b. How many sign changes are there in this new function? (1 point)

c. How many negative real roots are possible? (2 points)

Part IV: In light of the number of total roots for this function, how many possible complex roots are there? (2 points)

7.

Part I: What are the possible rational roots of this polynomial? (2 points)

Part II: Show your work in finding the first root of this polynomial. (4 points)

Part III: Use the resulting quadratic function to find the remaining 2 roots. (4 points)

8.

Part I: What are the possible rational roots of this polynomial? (2 points)

Part II: Show your work in finding the first root of this polynomial. (2 points)

Part III: Write the resulting cubic function. (2 points)

Part IV: Use synthetic division to find a second root that will reduce the cubic expression to a quadratic expression. (2 points)

Part V: Use any method you like to find the final 2 roots. (2 points)

9. Write a polynomial with rational coefficients having roots 3, 3 + i, and 3 -i.

Part I: Multiply the 2 factors with complex terms to produce a quadratic expression. (4 points)

Part II: Multiply the quadratic expression you just found by the 1 remaining factor to find the resulting cubic polynomial. (4 points)