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Algebra Review Exercises

1. Use long division to divide f(x) = x4-3x3+x-1 by g(x) = x2-x+1.

2. Use synthetic division to divide f(x) = x4 - 3x2 + x - 1 by x + 2.

3. Use the remainder theorem to find f(2) if
f(x) = x4 - 3x3 + 2x2 - 6x + 1:

4. Use the factor theorem to de termine whether x - 1 is a factor of
f(x) = x5 - 3x3 + x + 1.

5. Find the quotient and the remainder of f(x) = x4 - 3x + 1 divided
by x - 3.

6. Use synthetic division, and the fact that -3 is a root of f (x) =
x3 + 4x2 + x - 6 to ex press f (x) as a product of 3 linear factors.

7. List all the possible rational zeros of f(x) = 6x3 - 5x2 + 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) = 4x3 - 3x2 + 5x - 7.

9. Use Descartes’ Rule of Signs to determine the possible numbers of
positive and negative roots of f(x) = x5 - 4x3 + 3x + 1.

10. Use the general strategy developed in section 3.3 to find all the real
zeros of g(x) = 2x4 + x3 - 11x2 - 4x + 12.

11. Show that all the roots of h(x) = 10x3+51x2+98x+177 lie between
x = -6 and x = 1.

12. Show that f(x) = x3 + 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) = x3(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) = x4 + 2x3 + 6x2 + 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.
 

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