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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 determine 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 express 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

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