1. Use long division to divide f(x) = x^{4}-3x^{3}+x-1 by g(x)
= x^{2}-x+1.

2. Use synthetic division to divide f(x) = x^{4} - 3x^{2} + x -
1 by x + 2.

3. Use the remainder theorem to find f(2) if

f(x) = x^{4} - 3x^{3} + 2x^{2} - 6x + 1:

4. Use the factor theorem to determine whether x - 1 is a factor of

f(x) = x^{5} - 3x^{3} + x + 1.

5. Find the quotient and the remainder of f(x) = x^{4} - 3x
+ 1 divided

by x - 3.

6. Use synthetic division , and the fact that -3 is a root
of f (x) =

x^{3} + 4x^{2} + x - 6 to express f(x) as a product of 3 linear factors.

7. List all the possible rational zeros of f(x) = 6x^{3} -
5x^{2} + 4x - 15

determined by the Rational Zero Theorem .

8. Use the Upper and Lower Bound Theorem to determine upper and

lower bounds for the roots of f (x) = 4x^{3} - 3x^{2} + 5x - 7.

9. Use Descartes’ Rule of Signs to determine the possible
numbers of

positive and negative roots of f(x) = x^{5} - 4x^{3} + 3x + 1.

10. Use the general strategy developed in section 3.3 to
find all the real

zeros of g (x) = 2x^{4} + x^{3} - 11x^{2} - 4x + 12.

11. Show that all the roots of h(x) = 10x^{3}+51x^{2}+98x+177
lie between

x = -6 and x = 1.

12. Show that f(x) = x^{3} + 3x - 5 has a zero between x = 1
and x = 2.

Estimate this zero to within 1/4.

13. Determine the multiplicities of the zeros of f(x) =
x^{3}(x-1)^{5}(x+3)^{2}.

Then graph the function .

14. Find a polynomial f of degree 2 which has 1+i as a
zero and which

satisfies f(2) =-1.

15. Use the zero -1 + 2i of f(x) = x^{4} + 2x^{3} + 6x^{2} + 2x + 5 to find all

the zeros.

16. Use the graph of f (x) =1/x to graph

17. Find the horizontal asymptote of the function

.

18. Find the vertical asymptotes of

19. Determine whether
is even or odd.

20. Use the general strategy to graph the rational function

21. Suppose that a company has fixed costs of $2000 per
month and

each unit that it produces costs $10 to make. Find a formula for the cost

function C(x) to produce x units. Then find a formula for the average cost

per unit, and explain the meaning of the
horizontal asymptote of the

function.