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College Algebra 1110-005: Test #3

Welcome to the third College Algebra test! Your weapons of choice are a pen or pencil,
this test, and your brains. You are not al lowed to use any books, notes (except the provided
crib sheet), or calculators . Some general hints to help you maximize your points:

1. Show as much work as you can, since doing so gives me more opportunities to award
you with partial credit.

2. Don’t spend time converting your solutions to decimal notation — it’s perfectly acceptable
(in fact, it’ s preferred !) to write 1/3 instead of 0.333 . . . or ln(2) instead of
0.693 . . ..

3. By the same token, don’t spend too much time expanding polynomials. As a rule of
thumb, it’s perfectly acceptable to leave any polynomial with a degree higher than two
in factored form.

4. Be sure to read each question carefully! Failure to do so will result in a significant loss
of points, even if the answer you arrive at is correct.

5. If you have any questions during the test, don’t hesitate to ask me! If a problem is
unclear to you, it is probably unclear to your classmates as well.

Good luck, and may the Force be with you!

True & False

1. Mark the following statements as either true or false. If the statement is false, provide
a counter-example or justification.

(a) (2 points) Every polynomial of degree n has n factors (not all of which may be
distinct) over the complex numbers

(b) (2 points) If the complex number a + is a factor of the polynomial p(x), then so
is a −.

(c) (2 points) Every one-to- one function has an inverse.

(d) (2 points) Every exponential function of the form f(x) = ax, with a > 0, is a
strictly increasing function.

(e) (2 points) If ax = ay, then x = y.

(f) (2 points) The inverse of a logarithmic function is an exponential function.

Algebraic Manipulation

2. Solve each equation for x . Be sure to simplify your answer as far as possible, and convert
any logarithms in your final results to the natural logarithm (“ln”) if applicable.

Short Answer

3. (8 points) Factor the polynomial using long division and the Rational
Zeros Theorem . (Hint: The Rational Zeros Theorem states that a polynomial of the
form can only have rational zeros of the form p/q, where p
is a factor of a0 and q is a factor of an.)

4. (8 points) Write a fifth degree polynomial with real coefficients that has the roots r 1 = 0,

5. (8 points) Let

What is f1(x)?


6. Consider the exponential function f(x) = 22-x+ 1.
(a) (4 points) Draw a graph of f (x) in the space provided.

(b) (6 points) What is the domain of f(x)?

(c) (6 points) What is the range of f(x)?

(d) (4 points) What is f(0)?

(e) (4 points) For what value(s) of x does f(x) = 2?


7. Consider the case of continuously compounding interest .
(a) (8 points) For fixed principle P, interest rate r, and compounding period n, how
long will it take double your investment? (Hint: Your annuity is 2P in this case.
Solve for t.)

(b) (8 points) For fixed principle P, compounding period n, and time t, what interest
rate is required to double your investment? (Hint: Your annuity is again 2P, but
this time you must solve for r.)

See You Space Cowboys. . .

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