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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/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 nonzero. (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 (xaxis). 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 xaxis. 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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