Download Math 65, Exam 2 Review Name: 1. What is the definition of a

Math 65, Exam 2 Review
Name:
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1. What is the definition of a solution to a system of equations in two variables?
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2. Identify the slope and y-intercept of the following equations and then use them to solve the
system of equations by graphing. Remember to label the y-intercepts and the intersection of
the lines. Write your solution in set notation. You will earn NO points for solving this
symbolically.
3y − x = 6
3y = 4x − 12
y
x
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3. Solve the system by the addition method. Write the solution in set notation and show a
check.
3x − 5y = −21
−4x + 2y = 14
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4. Let x represent the first number and let y represent the second number. Suppose that twice
the first number, increased by 5 times the second number results in 12. Further assume the
first number equals − 52 of the second number, increased by 6. Use the given conditions to
write a system of equations and then solve that system using the substitution method. Write
the solution in set notation.
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5. Identify each polynomial as a monomial, a binomial or a trinomial. Give the degree of the
polynomial.
b. 324x3 − 9y 192 + 13
a. 15
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6. Add or subtract the following polynomials:
a. (15y 3 + 6y − 2) + (15y 2 − 2y 3 + 7)
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b.
2
3
x3 + 15 x −
3
4
− − 23 x3 − 34 x + 34
7. Multiply the following by first distributing one polynomial into the other and then
distributing a second time. i.e. the long way.
3x − 16 (12x3 − 18x2 + 30x − 24)
2
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8. Multiply the following by using the correct form when appropriate and the FOIL method or
the ”fast” multiplication when no form is present.
a. (−3x2 y 6 )(5x2 y 15 )
c. (3x − 4)(3x + 4)
b. (3x2 − 5y)2
d. (5x − 6)(2x2 + 3x − 4)
9. Simplify the following expressions by using the exponent rules gone over in class. Final forms
should have only positive exponents.
a. (−9xyz)0
c.
12z 3 −6z 2 +24z
6z
b.
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d.
−20x12 y −4
25x−3 y −5
15y 4
25x2
−3
10. Simplify the following expressions using scientific notation. Write the simplified form in
scientific notation.
a. (3 × 107 )(5 × 10−23 )
b. (2 × 103 )5
3
11. Simplify the following expressions involving radicals.
a.
b.
√
144 + 25
√
4
d.
3
√
3
−16
c.
e.
f.
1
125
g.
−1000
h.
72y 100
2x
9
·
9
2
400x10
10x3
341 x102 y 17
12. Rationalize and simplify the following expressions.
3
a. √
5
c.
2x
b. √
6
d.
4
x
√
4 2
6x2
√
8x−3 18
13. Factor the GCF out of the following polynomials.
a. 5x2 − 35x
b. 2x3 (a + 1) − 4x2 (a + 1) − 2x(a + 1)
14. Factor the following trinomials by unfoiling.
a. x2 − 5x − 14
b. y 2 + 3x − 10
15. Factor the following trinomials using the AC method.
a. 2x2 − x − 6
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b. 6x2 − 23x + 20
16. Factor the following special form polynomials.
a. 49z 2 − y 4
c. 27 − y 3
b. 16x2 + 40xy + 25y 2
d. 64x3 + 125
5
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17. Completely factor the following polynomials or state that they are prime.
a. 12x3 + 36x2 y + 27xy 2
d. y 7 + y
b. 6x2 − 6x − 12
e. 35w2 − 2w − 1
c. 16a3 b2 − 4ab2
f. 16x4 y − y 5
6
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18. Solving the following quadratic equations. State the solution(s) using set notation.
a. 0 = x2 + 5x + 6
c. 2x2 = −x + 6
b. 3x2 − 5x − 2 = 0
d. x(4x + 3) = 1
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19. The half-life of Uranium 234 is 2.5 × 105 years and the half-life of Plutonium is 8.0 × 107
years. How many times greater is the half-life of Plutonium than Uranium 234?
Solve the following story problems involving systems of equations by following these steps:
Step 1: Define variables to be the unknown quantities.
Step 2: Write a system of equations that model’s the problem’s conditions.
Step 3: Solve the system of equations.
Step 4: Write a conclusion which answers the story problem using complete sentences.
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1. A rectangular lot is being fenced on three sides. The fencing along the lot’s length costs $22
per foot while the fencing along the two side widths costs $8 per foot. The total fencing
comes out to 170ft while the cost of the fencing along the three sides comes to $2340. What
are the lot’s dimensions?
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2. By some magical means or via hard work you manage to save $25, 000 and decide to invest it
into two different stocks. At the end of the year you find that the first stock earns 1.2%
interest while the second stock earns 4% interest. Supposing you earn a total of $776 at the
end of the year, how much did you initially invest in each stock?
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3. A ball is launched straight up from a 300 foot high rooftop. The formula
h = −16t2 + 80t + 300
describes the ball’s height above the ground, h, in feet, t seconds after the ball was thrown.
The ball misses the rooftop on its way down and eventually strikes the ground.
a. How long will it take for the ball to hit
the ground?
b. When will the ball’s height be 204 feet?
4. A vacant rectangular lot is being turned into a community vegetable garden measuring 15
meters by 12 meters. A path of uniform width is to surround the garden. If the area of the
lot is 378 square meters, find the width of the path surrounding the garden.
x
Garden
2x + 12
12 meters
Path
x
15 meters
Path
2x + 15
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