MTH65 – Review for Final Exam
Name: ____________________________________
Note: On the final, you will be allowed to use a calculator, but no notes.
In problems 1-6, perform the indicated operation on the polynomials.
4 2
2 2
4 2
2
1.
(6x y – 9x y + 6xy) + (-5x y – 7x y – xy)
2.
(3x – x – 5x) – (2x – x + 3x – 9)
3.
(6x – 6)(5x + 3x + 1)
5.
x – 2x – 33x – 7
x–7
3
7
3
7
2
3
2
4.
(5y – 4)
6.
2x – 5x + 7x + 5
2x + 1
2
3
2
In problems 7-9, simplify the expression. Write your answers with only positive exponents.
7.
5
3 2
(4x )(-5x)(3x )
(6x )
12
x
-3
9.
 2x8 
 3
 3x 
12.
24x + 22x – 30x
-4 2
8.
In problems 10-12, factor the polynomial completely.
10.
xy – x + 7y – 7
11.
2
2
64x – 80xy + 25y
3
2
In problems 13-14, solve the equation using factoring.
13.
2
2x = 5x + 42
14.
(x – 2)(x + 4) = 7
2
15.
A ball is thrown straight up from a rooftop 288 feet high. The formula h = -16t + 48t + 288
describes the ball’s height, in feet, t seconds after it was thrown. How long will it take for the
ball to hit the ground?
16.
Find all numbers for which the expression
17.
Simplify the expression
x+6
is undefined.
x + x – 20
2
2
x – 8x + 16
2
x – 16
In problems 19-20, solve the rational equation.
18.
2y = 7 – 4
y+2
y+2
19.
5
7 – 4 =
x + 5 x + 1 x2 + 6x + 5
In problems 20-23, perform the indicated operation.
20.
10 y  2 y  3

y2  9 5 y2  y
2
21.
2
6x + 6y  x – y
7
x–y
2
22.
2
y + 4y – y – 8
2
2
y –y–6 y –y–6
23.
6y + 5
y – 36 y + 6
2
24.
To estimate the fur seal pup population, 6000 fur seal pups were tagged and released. A month
later, a sample of 700 pups were observed, and 222 of these were found to have been
previously tagged. Estimate the total number of fur seal pups. Round to the nearest whole
number.
25.
A tree casts a shadow 36 feet long. At the same time, a man 6 feet high casts a shadow 4.8 feet
long. How tall is the tree?
26.
The pitch of a musical tone varies inversely as its wavelength. A tone has a pitch of 660
vibrations per second and a wavelength of 1.6 feet. What is the pitch of a tone that has a
wavelength of 2.4 feet?
27.
The time required to assemble computers varies directly as the number of computers
assembled and inversely as the number of workers. If 30 computers can be assembled by 6
workers in 10 hours, how long would it take 5 workers to assemble 40 computers?
28.
Roy is designing a rectangular garden with a width of 8ft. The path that leads diagonally across
the garden is 2ft longer than the length of the garden. How long is the path?
29.
Solve
31.
If f ( x) 
1
2
3


for c.
ac ab bc
30. Find domain of the function f  x  
x2
x x2
2
2
2

find all x for which f(x) = 1
x 1 x  2
In problems 32- 36 use graph below to answer the questions.
32. Find the zeroes of the function.
33. Determine domain and range.
34. Determine x-intercepts.
35. Write the zeroes in factored form.
Write the original expression of the
function.
36. Given the graph below. Determine the vertical asymptote.
Answers
4 2
2 2
2
3
1.
x y – 9x y – 7x y + 5xy
3.
30x – 12x – 12x – 6
5.
x + 5x + 2 +
7.
-180x
8.
36
20
x
10.
(y – 1)(x + 7)
11.
(8x – 5y)
13.
 – 7 , 6


 2 
14.
{ 3, – 5}
15.
6 seconds
17.
x–4
x+4
18.

19.
{6}
20.
2
y(y + 3)
21.
6
7
22.
3
2
2
7
x–7
12
2.
x – 8x + 9
4.
25y – 40y + 16
6.
x – 3x + 5
2
2
2
9.
27
15
8x
12.
2x(4x – 3)(3x + 5)
-5, 4
11y – 30
(y + 6)(y – 6)
23.
24.
18,919 seal pups
25.
45 feet
27.
16 hours
28
17ft.
29.
c
30.
 x all real numbers, x  1, 2
32.
-3, 1
33.
Domain:  x all real numbers Range:  4,  
34.
(-3, 0), (1, 0)
35.
( x  3)( x  1)  x 2  2 x  3
36.
x=5
b  3a
2
26.
4
y–3
16.
440 vibrations per second
31.
-1, 4