Assignment 6

1. Explain why the product of two polynomials is a polynomial, but the quotient needn’t be.

2. Consider the function x³ – x + C for different values of C.

a. Sketch for C = -1, 0, and 1.

b. How many real roots does the equation have in each case?

c. What happens as C changes?

d. Why do we always have at least one real root?

3. Consider 2x⁴ + 11x³ + x² – 10x – 4.

a. What are the possible rational roots?

b. What are the actual rational roots?

c. Sketch. Are there other real roots?

d. Divide by (x-r) where r is a root. Repeat with as many roots as you know. Can you find the other roots?

4. Repeat the above with 4x⁴ + 8x³ – x² – 14x – 24.

5. Find quotient and remainder for

a.

b.