## C. HECKMAN TEST 2A

SOLUTIONS 170

(1) [10 points] For the rational function below, find its
x-intercept(s), y-intercept(s), vertical

asymptotes, horizontal asymptotes, and determine whether it is even, odd, or

neither. (If the function does not have any vertical asymptotes (for example),
make

sure you answer “none”; do not leave your answer blank.)

Solution: Its x- intercept is where y = 0, or when
, which is

when the numerator is zero . The numerator factors into 2(x − 2)(x − 3), so
the

x-intercepts are **2 and 3** .

The y-intercepts are where x = 0, so you just substitute x = 0 into the

formula to get or.

The vertical asymptotes occur where the denominator is zero but the numerator

isn’t. Since the denominator is never zero, there are** no vertical asymptotes** .

To find the horizontal asymptote, we look at the degrees of the numerator

and denominator. Since both degrees are equal, take the ratio of the
coefficients

of x^{2}: or** y = 2** .

Since , which is neither the original
function nor its

negative , this function is **neither** even or odd.

Grading: +2 points for each answer. Partial credit: +1 point for each answer.

2) Do the following for the equation 4^{z} = 19:

(a) [5 points] Convert it into logarithmic form.

Solution: .

Grading for partial credit: +2 points total for taking logarithms of both sides.

(b) [10 points] Solve for z , rounding your answer to **three decimal places.**

Solution: You can take logarithms of both sides here, and solve for z, or use
the

change of basis formula:

Note that will also
give you the correct answer.

(3) Solve the following equations for x . You must give **exact **answers!

(a) [10 points]

Solution: Since there are multiple logarithms, you need to combine them :

Now convert to exponential form :

[You got 4 points for getting this far.] If you
cross-multiply, you find out that

x + 37 = 10(x + 1) [7 points], so the only possible solution is **3** . It turns out
to

be an actual answer, after checking.

Grading for partial credit: +3 points for miscellaneous work.

(b) [10 points] 10^{2x} − 10^{x} − 12 = 0

Solution: If you let y = 10^{x}, you find out that

[3 points] so y = 4 or y = −3 [7 points]. To find x, you
solve the equations

10^{x} = 4 (which has a solution of log 4) and 10^{x} = −3 (which has no solutions).

So** log 4** is the only solution.

Grading for partial credit: +7 points for 4 or log(−3). Points were taken off

for an approximate answer.

(4) A bank offers a savings account where the interest is compounded 4.7%
monthly.

(a) [5 points] If you deposit $350 now, how much money will you have in 5 years?

Solution: Use the compound interest formula:

Grading: +2 points for the formula, +2 points for
substituting, +1 point for

the final answer.

(b) [10 points] When will you have $800 in your account?

Solution: You need to solve the equation

[3 points]

To solve this equation, you have to start by dividing both sides by 350; the

left-hand side is not 351.37^{12t}.

[7 points]

[10 points]

Grading for partial credit: −3 points for not dividing by 350 first.

(5) [10 points] Find a polynomial p (x) which has degree 3,
has 2 as a root, has −2 as a

root of multiplicity 2, and where p(0) = 16.

Solution: Since you know information about the roots (zeros), it is best to work

with the factored form. Since −2 has multiplicity 2, (x + 2)^{2} is a factor of p(x),

and (x−2) is also a factor of p(x). Note that if the multiplicity of 2 were
greater

than 1, then p(x) would have degree at least 4. Hence p(x) = A(x − 2)(x + 2)^{2}

for some constant A.

If you use the condition p(0) = 16, you find out that

so A = −2. Thus .

Grading for partial credit: +3 points (total) for involving (x − 2)(x + 2);

+7 points (total) for (x − 2)(x + 2)^{2}.

(6) [10 points] Find the quotient and remainder when 4x^{4}
+3x^{3} −4 is divided by x^{2} −2x.

Solution: Since the divisor is not of the form x − c, you must use long
division:

The quotient is , and
the remainder is .

Grading was done on a 0–3–5–7–10 point basis.

(7) [10 points] Using the Rational Root Theorem, list the
possible rational roots of the

polynomial 9x^{4} +5x^{3} −7x+5. You do **not** need to determine which are actual roots.

Solution: The Rational Root Theorem states that all rational roots of this
polynomial

are of the form ,
where p is a factor of 5, and q is a factor of 9. Since

the factors of 5 are 1, 5, −1, and −5, and the factors of 9 are 1, 3, 9, −1, −3,

and −9, we take all possible combinations to get the list below.

Grading: +3 points for the factors of 5, +3 points for the
factors of 9, and

+4 points for combining them. Grading for common mistakes: +8 points (total)

if the reciprocals were given.

(8) [10 points] The rational roots of the polynomial

are −2 and . Find all
roots of p(x) exactly, along with their multiplicities.

Solution: First of all, you should divide p(x) by x − (−2), and then divide that

quotient by , to try to find an equation that
the other roots

must satisfy. Doing this division (by long division or synthetic division)
yields

f(x) = 14x^{3} + 38x^{2} + 34x + 28.

You can’t use the quadratic formula on f(x), since f(x) has degree three.

The instruction to find the multiplicities suggests that −2 or
may have multiplicity

greater than one, so you should divide f(x) by x+2 and
again, to see

whether one of these divisions produces a remainder of 0. In fact, x+2 divides
into

f(x) without a remainder. The quotient of this final division is 14x^{2} + 10x +
14,

and we can use the quadratic formula to find the remaining roots:
.

The roots of p(x) are thus −2 (with multiplicity 2, since you divided by x+2

twice),, and
.

Grading: +5 points for finding f(x), +2 points for dividing again by x + 2,

+3 points for using the quadratic formula. Grading for partial credit: +5 points

(total) for −2 and .