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Lecture Guide Math 90 - Intermediate Algebra
to accompany
Intermediate Algebra, 3rd edition
Miller, O'Neill, & Hyde
Prepared by
Stephen Toner Victor Valley College
Last updated: 4/17/16
44
5.1 β Exponents & Scientific Notation
A. Properites of Exponents
1. π₯π β π₯π = π₯π+π
2. π₯π
π₯π = π₯πβπ
3. (π₯π)π = π₯πβπ
4. π₯0 = 1
5. π₯βπ =1
π₯π
*Simplify each of the following:
a. π₯4 β π₯8 =
b. π₯5 β π₯7 β π₯ =
c. 56 β 511 =
d. π₯14
π₯9=
e. π₯6π¦11π§14
π₯3π¦7π§12=
f. (2π₯2π¦15,000)0 =
g. 3π₯0 =
h. (3π₯)0 =
i. π₯β4
π₯2=
j. π₯3π¦β4
π₯β2π¦8=
Negative exponents are NOT considered
to be simplified. Do NOT leave them in
final answers!
45
k. (3
5)
2=
l. (3π₯2π¦3)4 =
m. (2π₯2π¦3)2(β3π₯π¦4)2 =
n. π₯β4
π₯β8=
o. 5β1
5=
p. 6β2 =
q. β6β2 =
r. β12π₯β4π¦β3
48π₯β7π¦5 =
s. Simplify: (3π₯β2π¦4
6π₯5π¦β7)β3
=
t. Simplify: (β3π’β3
π€β6 ) (β2π’2π£3π€2)β3
46
B. 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 10β5
*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. 4.5 x 10β4
1.5 x 1019
9. 2.4 x 108
4.8 x 1042
47
5.2 β Adding and Subtracting Polynomials
monomial
binomial
trinomial
polynomial
Vocabulary: ππ₯π + ππ₯πβ1 + β― + ππ₯ + π
*Given: 5π₯7 + 4π₯6 + 3π₯5 β― + 5π₯ β 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 5π₯2π¦7?
*Add: (3π₯2 + 5π₯ β 2) + (7π₯2 β 9π₯ + 13)
*Add:
5π₯3+3π₯2 +118π₯3β9π₯2+5π₯β3
*Find the perimeter:
*Subtract: (3π₯2 + 5π₯ + 11) β (π₯2 + 7π₯ β 4)
*Subtract:
5π₯3β2π₯2+4π₯β92π₯3+7π₯2β11π₯+8
6x - 4
6x - 4
5x+3
8x+2
8x+2
48
5.3 β Multiplying Polynomials
*Multiply each of the following:
1. (β3π₯4)(5π₯5)
2. 4π₯(3π₯ β 7)
3. 5π2π3π4(3ππ7 β 5ππ2π5)
4. (π₯ + 5)(π₯ β 3)
5. (2π₯ β 3)(3π₯ + 5)
6. (3π₯ β 5)(2π₯ + 4)
7. (5π₯ β 1)(π₯ + 8)
8. (4π₯ β 7)(4π₯ + 7)
9. (π + π)(π + π)
10. (3π₯ β 2)(π₯2 + 4π₯ β 7)
11. (5π₯ + 7)(3π₯ β 2)
12. (3π₯ β 2)2
13. (3π₯2π¦4)2
49
5.4 β Dividing Polynomials
A. Dividing by a monomial
Create separate fractions and then simplify
each separately.
1. 10π₯4π¦3+15π₯2π¦8
10π₯2π¦9
2. (8π₯3π¦ β 4π₯7π¦5 + 2π₯2π¦4) Γ· (4π₯π¦8)
B. Dividing by a non-Monomial
Use long division.
Recall... 512 Γ· 31
3. (π₯3 + 4π₯2 β 2π₯ + 8) Γ· (π₯ β 1)
4. 4π‘3+4π‘2β9π‘+3
2π‘+3
50
5. (π₯3 β 27) Γ· (π₯ β 3)
6. (π4 β π3 β 4π2 β 2π β 15) Γ· (π2 + 2)
Long division always works; synthetic division
only works when dividing by __________
factors (those without exponents).
*Divide using synthetic division:
(π₯3 + 2π₯2 + 4π₯ β 5) Γ· (π₯ + 1)
steps:
51
Divide: (π₯3 + 4π₯2 β 2π₯ + 8) Γ· (π₯ β 1)
Divide: (π₯3 + 64) Γ· (π₯ + 4)
Synthetic division can also be used to evaluate
polynomials:
If π(π₯) = π₯4 β 3π₯3 + 5π₯2 β 8π₯ + 17, find
π(β2) in two ways.
Use synthetic division to determine whether
π₯ + 4 is a factor of π(π₯) = π₯3 + 2π₯2 β 5π₯ β 6.
52
5.5 β Factoring (GCF and Grouping)
A. Factoring Out a Greatest Common Factor
*Factor each of the following completely.
1. 24π₯ β 36
2. 18π₯2 β 18π₯
3. 20π₯5π¦3π§2 β 24π₯2π¦5π§
4. 14π₯5π¦3 β 28π₯7π¦2 + 35π₯2π¦8
5. β12π₯3 + 4π₯2 β 9
B. Factoring by Grouping
6. π₯2(π₯ β 5) + 7(π₯ β 5)
7. 5π₯(π₯3 + 2) β 8(π₯3 + 2)
8. 3π + 3π + ππ + ππ
9. 8π€5 + 12π€2 β 10π€3 β 15
10. 2π + 3ππ¦ + ππ + 6π¦
11. 12π₯2 + 6π₯ + 8π₯ + 4
12. 6π2π + 30π + 2π2 + 10
53
5.6 β Factoring Trinomials
* Factor each of the following:
1. π₯2 + 10π₯ + 16
2. π₯2 β 3π₯ β 18
3. π₯2 + 6π₯ β 40
4. π2 β 12π + 11
5. π2 + 8π + 16
6. 7π¦2 + 9π¦ β 10
7. 8 + 7π₯2 β 18π₯
8. 12π2 β 5π β 2
9. 12π¦2 β 73π¦π§ + 6π§2
10. 36π₯2 β 18π₯ β 4
11. 12π2 + 11ππ β 5π2
12. 16π₯2 + 24π₯ + 9
13. 6π4 + 17π2 + 10
14. 3π¦3 β π¦2 + 12π¦
54
5.7 β Factoring - Special Cases
The Difference of Two Squares
*Factor each completely:
1. π₯2 β 49
2. π₯2 β 64
3. π₯2 β 25 4. π₯2 β 10
5. π₯2 β1
36 6. π₯2 + 25
7. π₯2π¦2 β 100π§2
8. π₯4 β 16
9. π₯8 β π¦8
10. π₯2 β 1 11. 25π₯2 β 16
12. 100π₯2 β 49π¦2
13. 25π₯2 β 100
14. π₯2 β 6π₯π¦ + 9π¦2 β 16
55
The Sum & Difference of Two Cubes
Memorize:
π3 + π3 = (π + π)(π2 β ππ + π2)
π3 β π3 = (π β π)(π2 + ππ + π2)
"SOAP" means....
*Factor each completely:
1. π₯3 + 125
2. π¦3 β 64
3. 8π₯3 β 1
4. 3π₯3 + 81
Factoring β General Strategy
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?
*Factor each of the following completely.
1. 2π2 β 162
2. 3π2 β 9π β 12
3. 64 + 16π + π2
4. 5π3 + 5
56
5. 3π¦2 + π¦ + 1
6. βπ3 β 5π2 β 4π
7. 14π’2 β 11π’π£ + 2π£2
8. 81π’2 β 90π’π£ + 25π£2
9. 12π₯2 β 12π₯ + 3
10. π‘4 β 8π‘
5.8 β Quadratic Equations and Word Problems
Quadratic equations are of the form
Zero Product Rule:
If π΄ β π΅ = 0, Then π΄ = 0 ππ π΅ = 0.
Solve each of the following equations.
1. (π₯ + 2)(π₯ β 3) = 0
2. (π₯ + 5)(2π₯ β 3) = 0
3. π₯2 β 2π₯ β 15 = 0
4. π₯2 β 8π₯ + 16 = 0
5. π₯2 β 24 = 2π₯
57
6. x2 β 25 = 0
7. 2x2 β 50 = 0
8. x3 β 25x = 0
9. 2x3 β 50x = 0
10. 4x2 β 11x = 3
11. 2m3 β 5m2 β 12m = 0
12. 2y2 β 20y = 0
13. 2y3 + 14y2 = β20y
14. 3x(x β 2) β x = 3x2 + 4
15. (x β 1)(x + 2) = 18
58
16. If a number is added to two times its
square, the result is 36. Find all such numbers.
17. The length of a rectangle is three times its
width. Find the dimensions if the area is 48
cm2.
18. A stone is dropped off a 256-ft. cliff. The
height of the stone is given by
β = β16π‘2 + 256, where t is the time (in
seconds). When will it hit the ground?
19. Use the Pythagorean Theorem to find x:
8
10 x
59
20. 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.
21. Write the quadratic equation whose
roots are β1 and 4, and whose leading
coefficient is 3.
22. A 17-foot ladder is leaning against a wall.
The distance between the base of the ladder
and the wall is 7 feet less than the distance
between the top of the ladder and the base of
the wall. Find the distance between the base of
the ladder and the wall.
23. Find the x- and y- intercepts of the
function 2
( ) 1 2 3f x x x x .
60
Chapter 5 Review
1. Divide: π₯3+64
π₯+4
2. Divide:
(5π₯3 + 10π₯2 β 15π₯ + 20) Γ· (15π₯3)
3. Simplify: (3π₯ β 2π¦)2
4. Add: (4π₯ + 2) + (3π₯ β 1)
5. Subtract 3π₯2 β 4π₯ + 8 from π₯2 β 9π₯ β 11.
6. Multiply: (3π₯ + 5)(2π₯ β 7)
7. Multiply: (π₯ β 4)(π₯2 + 5π₯ β 3)
8. The square of a number is subtracted from
60, resulting in β4. Find all such numbers.
9. The length of a rectangle is 1 ft. longer than
twice its width. If the area is 78 ft2, find the
rectangle's deimensions.
61
10. Factor: π₯2 + π₯ β 42
11. Factor: π4 β 1
12. Factor: β10π’2 + 30π’ β 20
13. Factor: π¦3 β 27
14. Factor: 49 + π2
15. Factor: 2π₯3 + π₯2 β 8π₯ β 4
16. Factor: 3π2 + 27ππ + 54π2
17. Solve (x β 2)(x + 5) = 44