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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/x2 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/x2 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
k4 + 3k2 + 1 = 0, but there are no real numbers that satisfy this equation. (8
pt)

5. = x4 + 3x3 - 24x2 - 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)(x3 - 3x2 - 6x + 8). We also know
from (A) that -2 is a root, so that means (x + 2) is another factor. So, divide
(x3 + 3x2 - 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) = x2-5x+4. Hence, = (x+6)(x+2)(x2-5x+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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