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Math III Fall Benchmark Test #2 Review
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1
 4 3
 3 4
  8 4
 2  3  5  1
A
B
C
D



1
1
2 
 2 5
 2  4  5
 1 7
4
2
1
 3 4 

E
 5 1 


 2  1
Find each of the following, if possible, using the matrices above.
1. A – C
2. AB
3. A-1
4. |C|
5. DE
6.
What is the multiplicative identity matrix for a 3 x 3 matrix? What is the additive identity matrix for a
3 x 3 matrix?
7.
If A, B, and C are matrices, give examples if the associative and commutative properties of addition
and multiplication. Which are true for matrices, and which are false?
8.
2 x  3 y  19
Solve using matrices: a) 
 7x  y  9
9.
 y  3x  1
Solve by graphing: 
 y  6x  5
10.
 4 x  5 y  11
b) 
3x  2 y  9
Find the maximum and minimum values of each function for the polygonal convex set determined by
given system of inequalities.
y  x  8

 4 x  3 y  3

a) 
x  8y  8

 f ( x, y )  4 x  5 y
 3x  2 y  0

y0

b) 
 3 x  2 y  24
 f ( x, y )  7 y  3 x
11.
The International Canine Academy raises and trains Siberian sled dogs and dancing French poodles.
Breeders can supply the academy with at most 20 poodles and 15 Siberian huskies each year. Each poodle
eats 2 lb of food per day and each dog eats 6 lb of food per day. Food supplies are restricted to at most 100 lb
per day. A poodle requires 1,000 hours per year of training whereas a sled dog requires 250 hours of training
per year. The academy cannot provide more than 15,000 hours per year of training time. If a poodle sells for
a profit of $200 and each sled dog sells for a profit of $100, how many of each king of dog should the
academy raise in order to maximize profits?
12.
Find the area of a triangle with the following vertices:
a) (-6, -3), (4, 12), (-1, 23)
b) (7, 21), (-32, -52), (81, 54)
13.
The senior classes at Luella and Woodland planned separate trips to Disneyworld. At Luella they
rented and filled 7 vans and 2 buses with 201 students. At Woodland they rented and filled 10 vans
and 6 buses with 460 students. How many can a van hold? A bus?
14.
Flying to Atlanta with a tailwind a plan averages 136 km/h. On the return trip the plan only averaged
128 km/h while flying back in the same wind. Find the speed of the wind and the speed of the plane
still in the air.
15.
Draw the vertex-edge graph that corresponds to the following matrix:
A B C D E
A 0 1 2 0 0 
B 1 0 1 0 0 
C  2 1 0 1 1
D  0 0 1 0 1


E  0 0 1 1 0 
16.
Write the matrix that corresponds to the vertex edge graph below.
17.
Use the Remainder Theorem to find the remainder when each polynomial is divided by the given
binomial. State whether the binomial is a factor of the polynomial.
a)
18.
x 5  x 3  x;
x3
b)
2 x 3  x 2  3x  7; x  2
Divide.
a) x 3  2 x 2  5x  1; x  1
b) 2 x 3  3x 2  8x  3;
x3
c) x 4  5x 3  14 x 2 ;
x2
19.
Find all possible rational roots, and then find the actual rational roots.
a)
20.
f ( x)  6x 3  5x 2  3x  2
f ( x)  8x 4  3x 3  5x 2  x  2
f ( x)  8x 4  3x 3  5x 2  x  2
25.
f ( x)  3x 3  2 x 2  x  1
b)
f ( x)  3x 3  6 x 2  9 x  5
Write the polynomial equation of least degree with the following roots:
a) 2, -1, 5
24.
b)
Describe the end behavior.
a)
23.
b) ( x 3  4 x 2  kx  1)  ( x  1)
Find the number of possible positive and negative real zeros.
a)
22.
f ( x)  x 3  2 x 2  8x
Find the value of k so that the remainder is zero.
a) ( x 3  kx 2  2 x  4)  ( x  2)
21.
b)
b) -3, 4i, -4i
Sketch the graph that indicates the shape of the polynomial function with the given characteristics.
a)
degree 7, positive leading coefficient, real zeros 4 (multiplicity of 3), -1 (multiplicity of 2),
and 2 (multiplicity of 2)
b)
degree 4, negative leading coefficient, real zeros -3 (multiplicity 1), 1 (multiplicity 2), and 5
(multiplicity 1)
Determine the possible number of positive and negative real zeros.
a)
f ( x)  3x 3  6 x 2  9 x  5
b)
f ( x)  x 5  7 x 4  4 x 3  3x 2  9 x  15