48
48
5.1, 5.2, 5.3 โ Properites of Exponents last revised 6/7/2014
Properites of Exponents
1. ๐ฅ๐ โ ๐ฅ๐ = ๐ฅ๐+๐
2. ๐ฅ๐
๐ฅ๐ = ๐ฅ๐โ๐
3. (๐ฅ๐)๐ = ๐ฅ๐โ๐
4. ๐ฅ0 = 1
5. ๐ฅโ๐ =1
๐ฅ๐
*Simplify each of the following:
a. ๐ฅ4 โ ๐ฅ8 =
b. ๐ฅ5 โ ๐ฅ7 โ ๐ฅ =
c. 56 โ 511 =
d. ๐ฅ14
๐ฅ9 =
e. ๐ฅ6๐ฆ11๐ง14
๐ฅ3๐ฆ7๐ง12=
f. (2๐ฅ2๐ฆ15,000)0 =
g. 3๐ฅ0 =
h. (3๐ฅ)0 =
i. ๐ฅโ4
๐ฅ2=
j. ๐ฅ3๐ฆโ4
๐ฅโ2๐ฆ8=
Negative exponents are NOT considered to be
simplified. Do NOT leave them in final answers!
49
49
k. (3
5)
2=
l. (3๐ฅ2๐ฆ3)4 =
m. (2๐ฅ2๐ฆ3)2(โ3๐ฅ๐ฆ4)2 =
n. ๐ฅโ4
๐ฅโ8=
o. 5โ1
5=
p. 6โ2 =
q. โ6โ2 =
r. โ12๐ฅโ4๐ฆโ3
48๐ฅโ7๐ฆ5 =
s. (3๐ฅโ2๐ฆ4
6๐ฅ5๐ฆโ7)โ3
=
6. Simplify: (โ3๐ขโ3
๐คโ6 ) (โ2๐ข2๐ฃ3๐ค2)โ3
Simplify each:
5๐ฅโ3 73 โ 711 5(๐ฅ2๐ฆ3)0
50
50
5.4 โ Scientific Notation
Scientific notationis a shorthand notation for
writing extremely small or large numbers.
Notation:
*Write each using scientific notation:
1. 9,374,000
2. 19.4 trillion
3. 0.000381
*Write each in standard form:
4. 4.71 x 108
5. 3.21 x 10โ5
*Multiply. Write your answers in scientific notation:
6. (3.5 x 1011) (4.0 x 1023
)
7. (2.45 x 1017) (3.5 x 1012
)
*Divide. Write your answers in scientific notation:
8. 12.5 x 10โ4
2.5 x 1019
9. 2.4 x 108
4.8 x 1042
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51
5.5 โ Adding and Subtracting Polynomials
monomial
binomial
trinomial
polynomial
Vocabulary: ๐๐ฅ๐ + ๐๐ฅ๐โ1 + โฏ + ๐๐ฅ + ๐
*Given: 5๐ฅ7 + 4๐ฅ6 + 3๐ฅ5 โฏ + 5๐ฅ โ 11, find
the following:
a. leading coefficient
b. constant term
c. degree of the second term
d. degree of the polynomial
If a term has more than one variable, its degree
is the ________ of its exponents.
*What is the degree of the expression 5๐ฅ2๐ฆ7?
*Add: (3๐ฅ2 + 5๐ฅ โ 2) + (7๐ฅ2 โ 9๐ฅ + 13)
*Add:
5๐ฅ3+3๐ฅ2 +118๐ฅ3โ9๐ฅ2+5๐ฅโ3
*Find the perimeter:
*Subtract: (3๐ฅ2 + 5๐ฅ + 11) โ (๐ฅ2 + 7๐ฅ โ 4)
*Subtract:
5๐ฅ3โ2๐ฅ2+4๐ฅโ92๐ฅ3+7๐ฅ2โ11๐ฅ+8
6x - 4
6x - 4
5x+3
8x+2
8x+2
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52
5.6 โ Multiplying Polynomials
*Multiply each of the following:
1. (โ3๐ฅ4)(5๐ฅ5)
2. 4๐ฅ(3๐ฅ โ 7)
3. 5๐2๐3๐4(3๐๐7 โ 5๐๐2๐5)
4. (๐ฅ + 5)(๐ฅ โ 3)
5. (2๐ฅ โ 3)(3๐ฅ + 5)
6. (3๐ฅ โ 5)(2๐ฅ + 4)
7. (5๐ฅ โ 1)(๐ฅ + 8)
8. (4๐ฅ โ 7)(4๐ฅ + 7)
9. (๐ + ๐)(๐ + ๐)
10. (3๐ฅ โ 2)(๐ฅ2 + 4๐ฅ โ 7)
11. (5๐ฅ + 7)(3๐ฅ โ 2)
12. (3๐ฅ โ 2)2
13. (3๐ฅ2๐ฆ4)2
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53
Mini-Review of Sections 5.1 โ 5.6
1. Simplify: โ5๐ฅโ3๐ฆ
10๐ฅ4๐ฆโ5
2. Multiply: (4.3 x 108) (3.0 x 1017
)
3. Divide: 0.6 x 10โ15
2.4 x 1011
4. Multiply: โ5๐ฅ2๐ฆ3๐ง(8๐ฅ๐ฆ๐ง5 โ 4๐ฅ๐ฆ2๐ง3)
5. Multiply: (3๐ฅ2 + 4)(2๐ฅ2 โ 7)
6. Multiply: (๐ฅ โ 3)(๐ฅ2 + 3๐ฅ + 9)
7. Multiply: (๐ฅ2 + 4๐ฅ โ 2)(๐ฅ2 + ๐ฅ โ 5)
8. Calculate the area of the larger rectangle in
two ways.
x 5
3
x
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54
5.7 โ Dividing Polynomials
A. Dividing by a monomial
Create separate fractions and then simplify
each separately.
1. 10๐ฅ4๐ฆ3+15๐ฅ2๐ฆ8
10๐ฅ2๐ฆ9
2. (8๐ฅ3๐ฆ โ 4๐ฅ7๐ฆ5 + 2๐ฅ2๐ฆ4) รท (4๐ฅ๐ฆ8)
B. Dividing by a non-Monomial
Use long division.
Recall... 512 รท 31
3. (๐ฅ3 + 4๐ฅ2 โ 2๐ฅ + 8) รท (๐ฅ โ 1)
4. 4๐ก3+4๐ก2โ9๐ก+3
2๐ก+3
55
55
5. (4๐ฅ3 + 5๐ฅ โ 12) รท (2๐ฅ โ 3)
6. (๐ฅ3 โ 27) รท (๐ฅ โ 3)
7. (๐4 โ ๐3 โ 4๐2 โ 2๐ โ 15) รท (๐2 + 2)
In Math 90, you will learn another method for
dividing called Synthetic Division. This process
will only work when dividing by a linear factor
(those without exponents). Problems like #7
above cannot be done using synthetic division
since there is an exponent in the divisor.
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56
6.1 โ Introduction to Factoring
Factoring is _______________________ .
Factoring is a ________________ process.
A. Factoring Out a Greatest Common Factor
*Factor each of the following completely.
1. 24๐ฅ โ 36
2. 18๐ฅ2 โ 18๐ฅ
3. 20๐ฅ5๐ฆ3๐ง2 โ 24๐ฅ2๐ฆ5๐ง
4. 14๐ฅ5๐ฆ3 โ 28๐ฅ7๐ฆ2 + 35๐ฅ2๐ฆ8
5. โ12๐ฅ3 + 4๐ฅ2 โ 9
B. Factoring by Grouping
6. ๐ฅ2(๐ฅ โ 5) + 7(๐ฅ โ 5)
7. 5๐ฅ(๐ฅ3 + 2) โ 8(๐ฅ3 + 2)
8. 3๐ + 3๐ + ๐๐ + ๐๐
9. 8๐ค5 + 12๐ค2 โ 10๐ค3 โ 15
10. 2๐ + 3๐๐ฆ + ๐๐ + 6๐ฆ
57
57
11. 12๐ฅ2 + 6๐ฅ + 8๐ฅ + 4
12. 6๐2๐ + 30๐ + 2๐2 + 10
Review
1. Multiply: (๐ฅ + 1)(๐ฅ + 2)(๐ฅ + 3)
2. Simplify: 53
5โ8
3. Simplify: (5๐ฅ3๐ฆ2)โ2
4. Simplify: (4๐ฅโ3๐ฆ
6๐ฅ๐ฆโ5)โ3
5. Multiply: (5.6 x 1014) (3.0 x 1022
)
6. Multiply: โ4๐ฅ2๐ฆ3๐ง(8๐ฅ๐ฆ5 โ 11๐ฅ2๐ง3)
7. Multiply: (5๐ฅ2๐ฆ โ 8๐ง)2
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58
8. Divide:
(16๐ฅ4๐ฆ2 + 20๐ฅ5๐ฆ โ 24๐ฅ3๐ฆ8) รท (6๐ฅ2๐ฆ7)
9. Divide: (๐ฅ3 + 6๐ฅ โ 7) รท (๐ฅ + 1)
10. Divide: (๐3 โ 64) รท (๐ โ 4)
11. Factor: โ8๐ฅ3 + 16๐ฅ2 + 20๐ฅ
12. Factor: 5๐ฅ2(๐ฅ + 7) โ 8(๐ฅ + 7)
13. Factor: 2๐ + ๐๐ + 3๐๐ฆ + 6๐ฆ
14. Factor: ๐ฅ๐ฆ โ ๐ฅ๐ง + 7๐ฆ โ 7๐ง
15. Factor: 4๐ฅ3 + 3๐ฅ2๐ฆ + 4๐ฅ๐ฆ2 + 3๐ฆ3
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59
6.2 โ Factoring Trinomials, part 1
* Factor each of the following:
1. ๐ฅ2 + 10๐ฅ + 16
2. ๐ฅ2 โ 3๐ฅ โ 18
3. ๐ฅ2 + 6๐ฅ โ 40
4. ๐2 โ 12๐ + 11
5. ๐2 + 8๐ + 16
6. ๐ค2 โ 7๐ค + 12
7. 12๐2 โ 96๐ + 84
8. ๐ฅ3๐ฆ3 โ 19๐ฅ2๐ฆ3 + 60๐ฅ๐ฆ3
9. โ2๐2 + 22๐ โ 20
10. 5๐ค2 โ 40๐ค โ 45
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60
6.3 โ Factoring Trinomials, part 2
pattern: ๐๐ฅ2 + ๐๐ฅ + ๐
*Factor each of the following completely.
1. 3๐ฅ2 + 13๐ฅ + 4
2. 7๐ฆ2 + 9๐ฆ โ 10
3. 8 + 7๐ฅ2 โ 18๐ฅ
4. 12๐2 โ 5๐ โ 2
5. 12๐ฆ2 โ 73๐ฆ๐ง + 6๐ง2
6. 36๐ฅ2 โ 18๐ฅ โ 4
7. 12๐2 + 11๐๐ โ 5๐2
8. ๐ฆ2 โ 6๐ฆ โ 40
9. 16๐ฅ2 + 24๐ฅ + 9
10. 6๐4 + 17๐2 + 10
11. 3๐ฆ3 โ ๐ฆ2 + 12๐ฆ
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61
6.4 โ Factoring Trinomials, part 3
The a-c Method
1. 6๐ฅ2 + 7๐ฅ โ 3
2. 9๐ฅ2 โ 12๐ฅ + 4
3. 16๐ฅ2 + 10๐ฅ + 1
4. 16๐ฅ2 โ 34๐ฅ โ 15
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6.5 โ The Difference of Two Squares and Perfect Square Trinomials
The Difference of Two Squares
*Factor each completely:
1. ๐ฅ2 โ 49
2. ๐ฅ2 โ 64
3. ๐ฅ2 โ 25 4. ๐ฅ2 โ 10
5. ๐ฅ2 โ1
36 6. ๐ฅ2 + 25
7. ๐ฅ2๐ฆ2 โ 100๐ง2
8. ๐ฅ4 โ 16
9. ๐ฅ8 โ ๐ฆ8
10. ๐ฅ2 โ 1 11. 25๐ฅ2 โ 16
12. 100๐ฅ2 โ 49๐ฆ2
13. 25๐ฅ2 โ 100
14. ๐ฅ2 โ 6๐ฅ๐ฆ + 9๐ฆ2 โ 16
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63
15. ๐ฅ2 + 8๐ฅ + 16 โ 25๐ฆ2
16. ๐ค2 โ 10๐ค + 25 โ 36๐2
17. 100 โ ๐ฅ2 โ 16๐ฅ๐ฆ โ 64๐ฆ2
6.6 โ The Sum & Difference of Two Cubes
Memorize:
๐3 + ๐3 = (๐ + ๐)(๐2 โ ๐๐ + ๐2)
๐3 โ ๐3 = (๐ โ ๐)(๐2 + ๐๐ + ๐2)
"SOAP" means....
*Factor each completely:
1. ๐ฅ3 + 125
2. ๐ฆ3 โ 64
3. 8๐ฅ3 โ 1
4. 3๐ฅ3 + 81
What if you forget these formulas?
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64
A General Stategy for Factoring
1. Can I factor out a _______________ ?
2. How many terms are there?
a. if four, try ____________________.
b. If three, try ____________________.
c. If two, try ______________________
or try _________________________.
3. Can I factor further?
The following exercises are from pages 470-471
of your textbook, which are printed on the next
page for your use.
#5. 2๐2 โ 162
#15. 3๐2 โ 9๐ โ 12
#25. 64 + 16๐ + ๐2
#10. 5๐3 + 5
#30. 3๐ฆ2 + ๐ฆ + 1
#35. โ๐3 โ 5๐2 โ 4๐
#45. 14๐ข2 โ 11๐ข๐ฃ + 2๐ฃ2
#55. 81๐ข2 โ 90๐ข๐ฃ + 25๐ฃ2
#65. 12๐ฅ2 โ 12๐ฅ + 3
65
65
66
66
6.7 โ Solving Equations Using the Zero Product Rule
Quadratic equations are of the form
Zero Product Rule:
If ๐ด โ ๐ต = 0, Then ๐ด = 0 ๐๐ ๐ต = 0.
Solve each of the following equations.
1. (๐ฅ + 2)(๐ฅ โ 3) = 0
2. (๐ฅ + 5)(2๐ฅ โ 3) = 0
3. (8๐ฅ โ 5)(7๐ฅ + 2) = 0
4. 1
3๐ง (๐ง โ
5
8) = 0
5. ๐ฅ2 โ 2๐ฅ โ 15 = 0
6. ๐ฅ2 โ 8๐ฅ + 16 = 0
7. ๐ฅ2 โ 24 = 2๐ฅ
8. ๐ฅ2 โ 25 = 0
9. 2๐ฅ2 โ 50 = 0
10. ๐ฅ3 โ 25๐ฅ = 0
11. 2๐ฅ3 โ 50๐ฅ = 0
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67
6.7 #26 ๐ฆ2 โ 7๐ฆ โ 8 = 0
6.7 #28 ๐ค2 โ 10๐ค + 16 = 0
6.7 #30 4๐ฅ2 โ 11๐ฅ = 3
6.7 #35 2๐3 โ 5๐2 โ 12๐ = 0
6.7 #38 4(2๐ฅ โ 1)(๐ฅ โ 10)(๐ฅ + 7) = 0
6.7 #46 2๐ฆ2 โ 20๐ฆ = 0
6.7 #48 9๐2 = 1
6.7 #49 2๐ฆ3 + 14๐ฆ2 = โ20๐ฆ
6.7 #62 3๐ง(๐ง โ 2) โ ๐ง = 3๐ง2 + 4
6.7 #69 (๐ฅ โ 1)(๐ฅ + 2) = 18
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68
6.8 โ Quadratic Word Problems
A. Number Problems
6.8 #10 If a number is added to two times its
square, the result is 36. Find all such numbers.
B. Consecutive Integer Problems
Review...
Consecutive Integers
Consecutive Even Integers
Consecutive Odd Integers
6.8 #14 The product of two even consecutive
integers is 48. Find all such numbers.
6.8 #16 The sum of the squares of two
consecutive integers is 9 less than 10 times their
sum. Find all such integers.
The product of two consecutive odd integers is
63. Find all such integers.
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69
The perimeter of a rectangle is 22 cm and its
area is 24cm2. Find the length and width of this
rectangle.
C. Area Problems
6.8 #18 The width of a rectangular picture frame
is 2 inches less than its length. The area is 120 in2.
Find its dimensions.
The length of a rectangle is three times its
width. Find the dimensions if the area is 48
cm2.
D. Height of a Projectile
6.8 #26 A stone is dropped off a 256-ft. cliff. The
height of the stone is given by โ = โ16๐ก2 + 256,
where t is the time (in seconds). When will it hit the
ground?
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70
E. The Pythagorean Theorem
Find x:
6.8 #42 The longer leg of a right triangle is 1 cm
less than twice the shorter leg. The hypotenuse is 1
cm more than twice the shorter leg. Find the length
of the shorter leg.
Review of Chapters 5 & 6
1. Simplify: (5๐ฅ2๐ฆ)(โ3๐ฅ4๐ฆ3)
2. Simplify: 5๐ฅ0
3. Simplify: 3โ3
4. Simplify: (๐ฅ5)3
5. Simlify: (3๐ฅโ2๐ฆ4
6๐ฅ5๐ฆโ2)โ3
6. Simplify: (โ4๐ฅโ5๐ฆโ3๐ง
8๐ฅโ7๐ฆ4๐งโ3 )โ4
8
10 x
71
71
7. Write 5.37 x 104 in standard form.
8. Write 365,000,000 in scientific notation.
9. Multiply: (3.5 x 1011) (4.0 x 1023
)
10. Divide: 6.0 x 104
8.0 x 1023
11. Divide: ๐ฅ3+64
๐ฅ+4
12. Divide:
(5๐ฅ3 + 10๐ฅ2 โ 15๐ฅ + 20) รท (15๐ฅ3)
13. Simplify: (3๐ฅ โ 2๐ฆ)2
14. Add: (4๐ฅ + 2) + (3๐ฅ โ 1)
15. Subtract 3๐ฅ2 โ 4๐ฅ + 8 from ๐ฅ2 โ 9๐ฅ โ 11.
16. Multiply: (3๐ฅ + 5)(2๐ฅ โ 7)
17. Multiply: (๐ฅ โ 4)(๐ฅ2 + 5๐ฅ โ 3)
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72
18. The square of a number is subtracted from
60, resulting in โ4. Find all such numbers.
19. The product of consecutive integers is 44
more than 14 times their sum. Find all such
integers.
20. The length of a rectangle is 1 ft. longer than
twice its width. If the area is 78 ft2, find the
rectangle's deimensions.
21. A right triangle has one leg that is 2 ft.
longer than the other leg. The hypotenuse is 2
ft. less than twice the shorter leg. Find the
lengths of all three sides of hte triangle.
73
73
22. Factor: ๐ฅ2 + ๐ฅ โ 42
23. Factor: ๐4 โ 1
24. Factor: โ10๐ข2 + 30๐ข โ 20
25. Factor: ๐ฆ3 โ 125
26. Factor: 49 + ๐2
27. Factor: 2๐ฅ3 + ๐ฅ2 โ 8๐ฅ โ 4
28. Factor: 3๐2 + 27๐๐ + 54๐2
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74
Additional Review (Chapters 5&6) โ If Time
1. Write 463,000,000 in scientific notation.
2. Multiply: (2.5 x 1017) (6.0 x 10โ4
)
3. Divide: 6.0 ๐ฅ 10โ4
8.0 ๐ฅ 1019
4. Simplify each:
a. 5๐ฅ0
b. (5๐ฅ)0
c. (โ2
3)
2
d. โ8โ2
e. (โ8)โ2
f. (โ3๐2๐3)4
g. โ5๐ฅโ2๐ฆ3
โ10๐ฅ4๐ฆโ5
h. (โ3๐ขโ2๐ฃ
๐คโ3 )โ2
โ (๐ข๐ฃโ2)
5. Given 5๐ฅ2 + 3๐ฅ7 โ 2๐ฅ + 8, state find each:
a. Leading coefficient: ___________
b. degree of the polynomial: ___________
6. Find the degree of the polynomial:
๐ฅ2๐ฆ4๐ง + 3๐ฅ๐ฆ7๐ง2 โ 2๐ฅ4๐ฆ๐ง3
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75
7. Add: (2๐ฅ โ 5) + (4๐ฅ + 8)
8. Combine:
(๐ฅ2 + 4๐ฅ โ 9) โ (2๐ฅ2 โ 3๐ฅ + 4) + (5๐ฅ2 โ 11)
9. Multiply each:
a. (3๐ฅ โ 5)(2๐ฅ + 7)
b. (4๐ฅ + 7)(4๐ฅ โ 7)
c. (3๐ฅ + 2๐ฆ)(4๐ฅ โ 9๐ฆ)
d. (5๐ฅ โ 8)2
e. (๐ฅ + 5)(๐ฅ2 โ ๐ฅ + 2)
10. Divide: 5๐ฅ2๐ฆ3โ12๐ฅ๐ฆ4+10๐ฅ4๐ฆ
8๐ฅ3๐ฆ2
11. Divide: (6๐ฅ2 โ 5๐ฅ + 4) รท (๐ฅ โ 3)
12. Factor: 5๐ฅ2 + 10๐ฅ + 100
13. Factor: 60๐ฅ๐ โ 30๐ฅ๐ โ 80๐ฆ๐ + 40๐ฆ๐
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76
14. Factor: ๐ฅ2 โ 3๐ฅ โ 28
15. Solve: ๐ฅ2 โ 3๐ฅ โ 28 = 0
16. Factor: 2๐ฅ2 โ ๐ฅ โ 6
17. Factor: ๐ฅ4 โ 16
18. Factor: ๐ฅ3 โ 27๐ฆ3
19. Solve: ๐ฅ2 = 8๐ฅ
20. The length of a rectangle is 2 cm more than
three times its width. Find the dimensions if
the area is 56 cm2.