# Algebra Exam Solutions

1.
The numerator doesn 't factor, so sim-

plify by polynomial division . This is also the best way to determine how h(x)

is a transformation of 1/x^{2} So when you perform polynomial division, one gets a

quotient Q(x) = -4 and a remainder R = +1. Thus we can write the original

poly as

Therefore
to get h(x), shift 1/x^{2} to the right 3 units and down 4

units. (5 pt)

The horizontal asymptote is
since the degree of the numerator is equal

to the degree of the denominator . The vertical asymptote is
since the

denominator is zero at this point, but the numerator is non-zero. (3 pt)

(2
pt)

2. From the statement of the problem, we must assume that
the relationship

between dress sizes is linear . So, I expect a linear equation : y = mx + b. Use

two points on the graph given in the problem statement: (4, 32), (12, 48).

A) So,
.
Then, as usual, use another point on the

graph to calculate the y -intercept b. So, y = 2x + b => 32 = 2(4) + b => 32 =

8 + b => b = 24. Hence,
. (3 pt)

B) f(x) = 2x + 24, so f(6) = 2(6) + 24 = 36, f(8) = 2(8) +
24 = 40, and

f(10) = 2(10) + 24 = 44. (3 pt)

C) Clearly, f(x) = 2x+24 is a line. Thus, the graph passe
the vertical-line test

and is 1 - 1. So, f must have an inverse. So, f(x) = 2x + 24 => y = 2x + 24 =>

(3 pt)

(3 pt)

C)Horizontal asymptote: y = 0 (x-axis). This tells you
that the amount of drug

is never negative . The whole graph starts at zero , increases rapidly right after

it is ingested, then slowly filters out of a person' s system . (3 pt)

D) By the graph, the percentage is highest at t = 1.12
hrs. This corresponds to

c(1.12) = .5590 ≈ 56% of drug in a person's bloodstream. (3 pt)

4. If there were real numbers k for which (x - k) is a
factor of the polyno -

mial given, then the remainder upon polynomial division would be zero. In

other words, k is a root or zero of the polynomial. So, one must have that

k^{4} + 3k^{2} + 1 = 0, but there are no real numbers that satisfy this equation. (8

pt)

5.
= x^{4} + 3x^{3} - 24x^{2} - 28x + 48

A)x = -2 is a root, since
= (-2)^{4} +3(-2)^{3} -24(-2)^{2} -28(-2)+48 = 0

(2 pt)

. B) By Synthetic division (with -6 as divisor), one finds that the remainder is

zero. So, yes (x + 6) is a factor of
.
(32 pt)

C) In order to do this, one could graph the original polynomial and look for all

places the graph crosses the x-axis. Also, by successive use of synthetic
division:

By (B), (x + 6) is a factor so
= (x + 6)(x^{3} - 3x^{2} - 6x + 8). We also know

from (A) that -2 is a root, so that means (x + 2) is another factor. So, divide

(x^{3} + 3x^{2} - 6x + 8) by (x + 2) to get the quotient polynomial...by synthetic
di-

vision, you should find that the remainder is zero. Also, that the quotient poly

Q(x) = x^{2}-5x+4. Hence,
= (x+6)(x+2)(x^{2}-5x+4). Now one can easily

factor this remaining Q(x). Therefore,
.

(4 pt)

**BONUS:**

where mo and c are constants so, solve for v .

Thus, the inverse is...

The inverse function f^{-1}(v) gives the velocity corresponding to a given rela-

tivistic mass m.

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