Description
In This collection, we will go deep into math.
This collection will help all math and high school students.
Study Set Content:
Luttrell 2012
101
Name: ______________________
Date: _____
7a – Simplifying Exponents
Simplif
y
each of the following e
x
pressions:
1. 3(-2)
2
2. (3∙2)
2
3.
2
7
3
2
2
( )
4.
(
)
3
6
2
3
5
x
x
5.
8 2006
1003 2006
1003
1002
(
)
(
)
6.
8 2
3
3
2 5
2
3
3
2
(
)
(
{
})
−
+
− +
7.
( )
( )
100
0
5
100
3
3
xy
x y
x
x
8. 8
y
+ (-7
y
)
2
9.
(
)
(
)
4
10 4
2
2
x
x
+
10.
zy
x
y
z
4
3
2
2
⋅
÷
(
)
11.
(
)
2
4
2
4
4
x
x
12.
6 2
3 2
148
150
⋅
⋅
13.
(
)
−
5
30
2
3
x
x
14.
zz y x
x
z
(
)
2
4
3
3
÷
⋅
15.
(
)
(
)
5
3
7
6
5
2
3
2
2
2
−
+
−
+ −
Luttrell 2012
102
Name: ______________________
Date: _____
7b – More Simplifying Exponents
Simplif
y
each of the following e
x
pressions:
1.
y y y
2
4
5
2.
−
(
)
x y
4
3
2
3.
z
z
12
6
4.
(
)
(
)
a b
a b
5
3
4
2
÷
5.
(
)
x
2
4
6.
13
26
a
a
a
⋅ ⋅
7.
(
)
2
3
2
t
8.
1
2
2
x y
xy
y
x
⋅
⋅
9.
(
)
−
x y
3
5
2
10.
4
2
4
a
b
÷
(
)
11.
(
)(
)
2
3
3
3
4
2
5
xy
x y
12.
(
)
2
2
1
0
3
x
y z
−
−
−
13.
3
12
2
2
4
3
a c
n
ac
n
÷
14.
3
4
15
12
3
2
3
4
5
2
x
y
x
y
y
x
÷
⋅
15.
(
)
3
4
2
x
y
−
−
Luttrell 2012
103
Name: ______________________
Date: _____
7c – Prime Numbers
Prime numbers
are any number that cannot be divided evenly by another number except for one and itself.
1. Shade any number that is NOT a prime number in the list below:
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
Here are a few tricks by which you can tell whether a number is divisible:
•
Divided by 2: all even numbers.
•
Divided by 3: if the sum of the digits is divisible by 3.
•
Divided by 5: number ends with 0 or 5
•
Divided by 6: even and rule for 3 works.
•
Divided by 9: if sum of digits is divisible by 9.
•
Divided by 11: if a 3-digit number has the first and last digit sum to be the middle digit.
Using the number tricks above, fill in the blanks.
2. 22,245 is divisible by ____ and ____.
3. 473 is divisible by _____.
4. 6561 is divisible by _____ and _____.
5. 792 is divisible by 2, 3, ____, _____, and ____.
Luttrell 2012
104
3 2 *3 2*2
Name: ______________________
Date: _____
7d – Prime Factorization
Often times it is necessar
y
to break down an item into smaller pieces, whether it be a digestive
s
y
stem, rearranging a postal package contents, troubleshooting a computer problem, or an
y
other
instance. Working with numbers, the
factor
is a number that divides into another evenl
y
. For
e
x
ample, the factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24. A factor tree is a usual algorithm for
finding the factors of a number.
(American) Factor Tree:
(European) Factor Chart
18 - looks like long division
18
2 9 2 9
3 3
3 3 3 1
The usefulness of a factor tree is that
y
ou have divided up the number (18) into its smallest
factors. Those factors of 2 and 3 are onl
y
divisible b
y
1 and itself. That makes those factors
prime
. One is not considered prime; it is
unique
. While the goal of the factor tree is to get
prime factors, the other factors can be found by combining the different primes. The
prime
factorization
is a list of all the prime factors in ascending order. 18 would have a prime
factorization of 2×3×3 or 2×3
2
. The prime factorization of 24 is 2
3
×3 because 24 = 2×12 =
2×2×6 = 2×2×2×3. By combining different prime numbers of 24, you can get the entire factors
of 24: 2, 3, 4, 6, 8, 12, and 24. The factors of 18 are 2, 3, 6, and 9. Factoring can also take a
pol
y
nomial and split it into smaller pol
y
nomials. This idea that pol
y
nomials like
x
2
- 2
x
- 3 can
be factored into
x
- 3 and
x
+ 1 will be developed later.
When adding fractions together,
y
ou need to get a common denominator. Find the Least
Common Multiple (LCM) in order to reduce the work. For 18 and 24 the least common multiple
would be 72. There are two methods for finding the LCM. A common elementar
y
method is to
list the multiples of each number until
y
ou find a common number between the two.
18
36
54
72
90
108
126
144
162
180
198
216
24
48
72
96
120
144
168
192
216
240
264
288
Another method is to use the greatest power of each prime in the prime factorization. The prime
factorization of 18 is 2×3
2
and 24 is 2
3
×3, so the LCM would be 2
3
×3
2
which is 8 ×9 = 72. A
good wa
y
to visualize the result is to use a Venn diagram of the prime factors. Place the prime
factors of 18 in the left circle and the prime factors of 24 in the right circle. The factors in
common should be placed in the shared region. Now take each part of the circles and multiply
the factors together and you get 3∙2∙3∙2∙2 = 72.
Note: GCF is the shared region. This will be explained later.
Luttrell 2012
105
Name: ______________________
Date: _____
7d – Prime Factorization continued
Whereas the LCM, least common multiple, is the
union
of the two circles in a Venn diagram, the
GCF would be the
intersection
. The GCF, short for Greatest Common Factor (Divisor), is the
largest factor that divides into each number evenly. In the Venn diagram, it is the region
overlapped by both circles. So the GCF(18, 24) is 6.
Another wa
y
of finding the GCF is to write the prime factorization of each number and take onl
y
the primes that are in common, and with the least e
x
ponent. The prime factorization of
18 is 2×3
2
and 24 is 2
3
×3, so the GCF would be 2
1
×3
1
which is 6. Of course the way most
elementar
y
and middle schools teach is to list the factors of each number and take the greatest
common number: 18 is 1, 2, 3, 6, 9, 18 and 24 is 1, 2, 4, 6, 8, 12, 24.
The GCF is useful in simplif
y
ing equations. Say you had an equation such as 4
x
2
+ 12
x
- 20 = 0.
The greatest common factor of 4, 12, and 20 is 4. So you could divide both sides by 4 to get
x
2
+ 3
x
- 5 = 0, making it easier to solve.
The word factoring can be used in other manners.
Factoring out
implies dividing the GCF from
an e
x
pression. The e
x
ample above would be written as 4
x
2
+ 12
x
- 20 = 4(
x
2
+ 3
x
- 5). The
formal name is Converse to Distribution. Another e
x
ample of
factoring out
is 2
x
- 4
y
= 2(
x
- 2
y
).
Find the prime factorization of each:
1. 116
2. 175
3. 216
4. 40
Find the Lowest Common Multiple and Greatest Common Factor of each. Label.
5. 35 and 21
6. 18 and 42
7. 21 and 54
8. 3, 12, and 20
9. 24, 42
10. 15, 36
11. 35, 25, 50
12. 12, 28, 32
13. 64, 32, 56
14. 10, 42, 72
15. 36, 72, 84
16. 8, 15, 20
Luttrell 2012
106
Name: ______________________
Date: _____
7e –Simplifying Radicals
One of the properties of e
x
ponents that was not discussed before was that of fractional e
x
ponents.
As you can tell in the e
x
ample below, fractional e
x
ponents are another way of writing radicals.
x
x
a
b
a
b
=
81
81
3
27
3
4
3
4
3
=
=
=
Before manipulating radicals,
y
ou’ll need to understand the pieces to the radical. In the
e
x
pression
x
a
b
the
b
is the
root index
. It says how many of the same number is being
multiplied together to get
x
a
. The √ is the radical sign; it implies what operation needs to be
performed. The line over the
x
a
is the
vinculum
; it is a fanc
y
name for parentheses. So
4
can
be reduced to 2 because the same two numbers that multipl
y
to get 4 is 2. Note that for square
roots, the root inde
x
is usuall
y
dropped.
144
reduces to 12. But what happens when the
number is not a perfect square? You simplif
y
the
radicand
(e
x
pression inside the radical) so it
contains no factors that are perfect squares. For n
th
roots, you want no factors that are n
th
powers.
E
x
ample A:
24
4 6
4
6
2
6
2 6
2
2
=
⋅ =
⋅
=
=
E
x
ample B:
146
2 73
146
=
⋅
=
E
x
ample C:
108
4 27
4 9 3
2 3 3
6 3
=
⋅
=
⋅ ⋅ = ⋅
=
E
x
ample D:
56
2 28
2 2
2 7
2 14
=
⋅
=
⋅ ⋅ ⋅ =
(
)
Another way to simplify a radicand is to make a factor tree and look for pairs. Better yet is to
use prime factorization.
Simplif
y
e
x
actl
y
the following:
1.
24
2.
75
3.
96
4.
102
5.
144
6.
225
7.
625
8.
525
Luttrell 2012
107
Name: ______________________
Date: _____
7f –More Rational Exponents
Given:
x
x
a
b
a
b
=
Examples of rewriting fractional exponents:
16
16
16
4 4
4
1
2
2
=
=
=
⋅ =
27
27
3 3 3
3
1
3
3
3
=
=
⋅ ⋅ =
81
81
3 3 3 3
3
1
4
4
4
=
=
⋅ ⋅ ⋅ =
For rational exponents, most times it’s easier to simplify with the denominator of the exponent
first. See the examples below:
4
2
8
3
2
3
=
=
16
2
8
3
4
3
=
=
625
5
125
3
4
3
=
=
Simplify the following without a calculator into an integer or a simplified radical:
1.
64
2
3
2.
64
7
6
−
3.
64
5
6
4.
(
)
−
64
2
3
5.
−
−
32
6
5
6.
343
2
3
7.
−
−
100
3
2
8.
(
)
256
625
3
4
−
9.
−
−
125
27
4
3
b g
10.
6
2
÷
11.
81
9
3
3
⋅
12.
128
32
3
⋅
13.
64
3
14.
36
3
15.
64
4
5
3
Luttrell 2012
108
Name: ______________________
Date: _____
7g – Combining Radicals
Adding
radicals is similar to adding like terms. Like terms are combined by adding the
coefficients. When radicals have the same radicands and root indices, then add the coefficients.
E
x
ample A:
2 3
4 3
6 3
+
=
E
x
ample B:
2 3
4 3
3
+
Example B cannot be simplified because the root indices are different.
Simplify e
x
actl
y
:
1.
−
+
3 24
2 54
2.
−
+
3 3
2 27
3.
5 3
48
+
4.
3 1
4
−
5.
2
0
3
4
x
x
−
6.
8
2 32
+
Multiplying radicals can onl
y
be done easil
y
with those of the same root inde
x
. If the root
indices are the same, multipl
y
the radicands. Those with differing indices will be dealt with
later.
E
x
ample C:
2 24
48
4 3
=
=
E
x
ample D:
(
)
(
)(
)
2 5
2 5 2 5
4 5
20
2
=
= ⋅ =
See a trick? Share it!___________________________________________________
Simplif
y
each e
x
pression:
7.
6 12
8.
2 8 4
9.
(
)
3 6
2
10.
(
)(
)
3 8 2 12
11.
3
8
2
(
)
−
12.
(
)
5 3
2
Luttrell 2012
109
Name: ______________________
Date: _____
7h – Rationalizing Denominators
Dividing by radicals could require a calculator. But before those were invented, mathematicians
used a trick to get the radical out of a denominator. What’s the purpose? Dividing by a never-
ending number is quite impossible to do! But to divide by an integer is not bad at all! Even with
the invention of the calculator, most people prefer the answer to be written with the radical on in
the numerator. Mathematicians don’t consider a fraction with radicals simplified until it has
been rationalized.
The whole process of rationalizing the denominator is to multiply by one. That way the value
doesn’t change, but its appearance does. Write in
y
our own words what occurs in each step.
E
x
ample:
2
3
2
3
=
Definition of division
2
3
3
3
⋅
_______________________________
6
9
_______________________________
6
3
_______________________________
Simplif
y
:
1.
6
2
2.
3
5
3.
5
6
4.
2
6
5.
2
8
6.
4
2 3
7.
3
4 8
2
8.
2
56
Luttrell 2012
110
Name: ______________________
Date: _____
7i – Solving Radical Equations
Solve the following equations:
1. 2
x
2
-32 = 0
2. 3
x
2
= 75
3. 4
x
3
= 32
4.
x
− =
8
4
5.
x
+ =
5
20
6.
x
+ =
6
6
7.
x
− =
1
7
8.
x
+ =
3
5
9. 5x
4
=625
Luttrell 2012
111
Chapter 7 Test
Name: ___________________________
Date: ____________
SHOW WORK
. A calculator is allowed on this test. Attach any scratch paper that’s used.
1. With the numbers 16 and 24, find the
A. Greatest Common Factor.
B. Least Common Multiple.
2. With the numbers 18, 24, and 28, find the
A. Greatest Common Factor.
B. Least Common Multiple.
3. Write as a product and then evaluate:
A.
2
3
3
bg
B.
− −
4
4
b g
4. Write each product as a single power and then evaluate:
A.
(
)
−
⋅ −
−
3
3
3
4
b g
B.
7 3
3
3
⋅
5. Simplify each quotient:
A. 4
3
×5
2
÷ (4
2
× 5)
B. x
3
y
7
÷ (x
4
y
3)
6. Simplify each expression:
A. 8
5
× 8
-11
÷ 8
-3
B. x
3
(x
3
)
4
÷ (x
2
)
0
5
5
5
5
5
5
Luttrell 2012
112
Chapter 7 Test, continued
7. Simplify into scientific notation
A. 0.000 000 000 034
B. 3×10
8
+ 4.7×10
9
8. Solve for x:
a. 4
x
= 64
B. (125)
⅔
= x
2
C.
x
+ =
8
6
9. Short answer:
A. How can you tell if a radical (square root, cube root...) is a rational number?
B. Simplify: 4√3 + (3 - √3)
10. Simplify:
A.
18
6
B. 3 27 6
Bonus: (3pt) Let 10
101
-1 be written as an integer in standard form. What is the sum of the digits?
(2pt) Suppose N
1982
= 1982
1982
. If N ≠1982, what is the real value of N?
5
5
10
5
0
Luttrell 2012
113
Chapter 8 Expanding & Solving Polynomials
Luttrell 2012
114
Name: ______________________
Date: _____
8a – Multiplying Polynomials
If the teacher were to ask the class to multipl
y
2
x
+ 5 to
x
- 3, many students will give the
incorrect answer of 2
x
2
- 15. These students didn’t distribute correctl
y
, causing them to lose the
middle terms. There are four methods to ensure a complete answer: traditional distribution, box
method, FOIL, or multipl
y
ing like you did with real numbers. The letters of FOIL stand for the
product of first terms, outer terms, inner terms, and last terms of each binomial. The box method
uses a multiplication table or a Punnett Square (see biolog
y
class) with the binomials. The
answer for both methods is the sum of these products.
Traditional distribution
: (2
x
2
+ 4)(
x
- 1) = (2
x
2
+ 4)
x
+ (2
x
2
+ 4)(-1) = 2
x
3
- 2
x
2
+ 4
x
- 4
Box Method
:
x
-1
2
x
2
2
x
3
-2
x
2
4
4
x
- 4
F O I L
Multiply like Reals
:
(2
x
+ 5)(
x
- 3) = (2
x
)(
x
) + (-3)(2
x
) + (5)(
x
) + (5)(-3)
2
x
+ 5
= (2
x
2
) + (-6
x
) + (5
x
) + (-15)
×
x
- 3
= 2
x
2
- 6
x
+ 5
x
- 15
-6
x
- 15
= 2
x
2
-
x
- 15
2
x
2
+ 5
x
2
x
2
-
x
- 15
E
x
pand.
1. (
x
- 3
y
)(
x
+ 2
y
)
2. (
x
+5)
2
3. (
x
-2)(3-
x
)
4. (4
x
-3)(4
x
+3)
5. (2a+3b)(a-5b)
6. (
x
+ 2)(10 - 4
x
)
7. (W + 2)(W - 2) 8. (a - 4)(a + 4)
9. (11-
x
)(
x
+3)
10. (3
x
2
- 4)(
x
+ 2)
11.(
x
2
+ 2)(2
x
2
+ 1) 12. (2
x
2
- 6)(
x
2
+ 4)
Luttrell 2012
115
Name: ______________________
Date: _____
8b – Factoring
Factoring when
a
= 1,
c
> 0:
A quadratic e
x
pression of
ax
2
±
bx
+
c
where
a
= 1 looks like
x
2
±
bx + c
. This e
x
pression can be
factored into (
x+d
)(
x+ e
) or (
x-d
)(
x-e
) whenever the
c
term divisors sum up to the middle term
b
.
Note how both factors have the same sign as
b
whenever
c
>0.
E
x
ample A:
x
2
+ 5
x
+ 6
Y
ou probabl
y
won’t have more than one line of
( )( )
work on
y
our paper. But these are the steps to fill
(
x
)(
x
)
in the parentheses. Note (
x
2)(
x
3) was chosen
Choices: (
x
1)(
x
6) Or (
x
2)(
x
3) because 2 + 3 = 5. The signs are usuall
y
the last.
Answer: (
x
+ 2)(
x
+ 3)
E
x
ample B:
x
2
- 12
x
+ 36
( )( )
Fill in the missing numbers!
(
x
)(
x
)
Choices: (
x
±
)(
x
3) or (
x
1)(
x
36) or (
x
2)(
x
18) or (
x
4)(
x
±
) or (
x
6)(
x
±
)
Answer: (
x
- 6)(
x
- 6)
Factor. Check work.
1.
x
2
+ 4
x
+ 3
2.
x
2
- 2
x
+ 1
3.
x
2
+ 6
x
+ 8
4.
x
2
+ 7
x
+ 12
5.
x
2
- 14
x
+ 24
6.
x
2
- 11
x
+ 24
7.
x
2
+ 9
x
+ 18
8.
x
2
- 10
x
+ 24
9.
x
2
+ 11
x
+ 18
10.
x
2
+ 9
x
+ 20
11.
x
2
+ 13
x
+ 40
12.
x
2
- 5
x
+ 4
Luttrell 2012
116
Name: ______________________
Date: _____
8c – Factoring
Factoring when
a
= 1,
c
< 0:
A quadratic in the form of
x
2
±
bx
-
c
can be factored into (
x-d
)(
x+e
) whenever the
c
term
divisors differ b
y
the middle term
b
. The larger factor will have the same sign as
b
; the other
factor will have the opposite sign.
E
x
ample A:
x
2
+ 5
x
- 6
Y
ou probabl
y
won’t have more than one line of
( )( )
work on
y
our paper. But these are the steps to fill
(
x
)(
x
)
in the parentheses. Note (
x
6)(
x
1) was chosen
Choices: (
x
1)(
x
6) Or (
x
2)(
x
3) because 6 - 1 = 5. The signs are last to be done.
(
x
+ 6)(
x
- 1)
E
x
ample B:
x
2
-
x
- 30
( )( )
Fill in the missing numbers!
(
x
)(
x
)
Choices: (
x
±
)(
x
3) or (
x
1)(
x
30) or (
x
2)(
x
18) or (
x
5)(
x
±
)
(
x
- 6)(
x
+ 5)
Factor. Check work.
1.
x
2
+
x
- 6
2.
x
2
+ 3
x
- 10
3.
x
2
- 8
x
- 20
4.
x
2
+ 4
x
- 12
5.
x
2
- 6
x
- 16
6.
x
2
-
x
- 12
7.
x
2
- 16
8.
x
2
+ 3
x
- 18
9.
x
2
- 7
x
- 18
10.
x
2
- 36
11.
x
2
+ 8
x
- 9
12.
x
2
- 9
x
- 36
Luttrell 2012
117
Name: ______________________
Date: _____
8d – Factoring
Factoring when
a
≠ 1:
A quadratic e
x
pression of
ax
2
±
bx
±
c
can be factored in the same wa
y
as lessons 28a and 28b.
There is one e
x
tra step, and that is factoring the
a
term as well. This can create man
y
combinations to tr
y
. But after lots of practice,
y
ou will start to recognize patterns. Ask for one.
E
x
ample A: 4
x
2
- 4
x
- 3
With an
a
≠ 1 term, it just means
( )( )
there are more combinations to tr
y
.
(
x
)(
x
)
Choices: (2
x
1)(2
x
3) Or (4
x
1)(
x
3) Or (4
x
3)(
x
1)
Answer: (2
x
+ 1)(2
x
- 3)
E
x
ample B: 2
x
2
- 11
x
- 30
( )( )
Fill in the missing numbers!
(
x
)(
x
)
Choices: (2
x
±
)(
x
3) or (2
x
3)(
x
10) or (2
x
2)(
x
15) or (2
x
15)(
x
±
)
(2
x
1)(
x
±
) or (2
x
30)(
x
1) or (2
x
±
)(
x
6) or (2
x
5)(
x
6)
Answer: (2
x
- 15)(
x
+ 2)
Factor. Check work.
1. 2
x
2
+ 5
x
- 7
2. 3
x
2
- 12
x
- 15
3. 3
x
2
- 8
x
+ 4
4. 4
x
2
-
x
- 5
5. 3
x
2
- 11
x
+ 6
6. 4
x
2
- 8
x
+ 3
7. 3
x
2
-
x
- 4
8. 3
x
2
- 4
x
- 4
9. 8
x
2
- 6
x
- 9
10. 6
x
2
- 5
x
- 6
11. 6
x
2
+ 15
x
+ 6
12. 6
x
2
- 37
x
+ 6
Luttrell 2012
118
Name: ______________________
Date: _____
8e – Solve by Factoring
Once you know how to factor, then you are able to solve many quadratic equations for the
variable. Most equations will follow the steps as in the e
x
ample below.
E
x
ample: 2
x
2
=
x
2
+ 5
x
- 6
2
x
2
-
x
2
- 5
x
+ 6 = 0
Bring terms to one side of the equation.
x
2
- 5
x
+ 6 = 0
Simplif
y
(
x
- 2)(
x
- 3) = 0 Factor
x
- 2 = 0 Or
x
- 3 = 0
If
ab
= 0, then either
a
= 0 or
b
= 0.
x
∈
{2, 3}
Solve each equation for the variable.
E
x
ample: 0 = 2
x
2
+ 6
x
- 20
Fill in the blanks!
0 = 2(
x
2
+ 3
x
-
±
)
0 = 2(
x
+
±
)(
x
-
±
)
x
+
±
= 0 Or
x
- 2 = 0
x
∈
{2, -5}
Solve b
y
factoring:
1.
x
2
- 7
x
+ 6 = 0
2.
x
2
- 3
x
+ 2 = 0
3.
x
2
- 7
x
+ 12 = 0
4.
x
2
- 3
x
= 4
5.
x
2
+ 5
x
= - 4
6.
x
2
+ 13
x
+ 12 = 0
7.
x
2
+ 5
x
= 14
8.
x
2
- 8
x
+ 12 = 0
9.
x
2
+ 9
x
= -14
10. 2
x
2
+
x
= 6
11. 4
x
2
+ 9
x
= -5
12. 12
x
2
-
x
= 20
Luttrell 2012
119
Name: ______________________
Date: _____
8f – Difference of Squares
When a quadratic e
x
pression of
ax
2
±
bx
±
c
is missing the b
x
, it probably still can be factored.
As you noted before in a previous lesson,
x
2
-
c
can factored into (
x
- d)(
x
+ d). Then it can be
concluded
c
is a perfect square. The factors of c are equal, so when foiling d
x
- d
x
= b
x
= 0.
E
x
ample A:
x
2
- 49 = (
x
- 7)(
x
+ 7).
Check work:
x
2
-7x +7x
- 49
can be simplified to
x
2
- 49.
A leading coefficient must be factored into equal quantities as well.
E
x
ample B: 9
x
2
- 100 = (3
x
- 10)(3
x
+ 10)
Ma
y
be
y
ou can simplif
y
before factoring
E
x
ample C: 25
x
2
- 100 = 25(
x
2
- 4) = 25(
x
- 2)(
x
+ 2)
Summarize the following factor rules:
Perfect Trinomial
a
2
x
2
± 2ab
x
+
b
2
=
Difference of Squares
a
2
x
2
-
b
2
=
Solve b
y
factoring:
1.
x
2
- 9 = 0
2.
x
2
= 16
3.
x
2
- 25 = 0
4.
x
2
- 36 = 0
5. 25
x
3
- 100
x
= 0
6. 2
x
3
- 32
x
= 0
7. 3
x
2
- 27 = 0
8. 4
x
2
- 36 = 0
9. 25
x
2
- 9 = 0
Luttrell 2012
120
Name: ______________________
Date: _____
8g – More Practice Factoring
Use the following formulas to help factor the questions below.
a
2
– b
2
= (a – b)(a + b)
a
3
– b
3
= (a – b)(a
2
+ ab + b
2
)
a
3
+ b
3
= (a + b)(a
2
– ab + b
2
)
Factor:
1. 25x
2
– 9
2. 4 – 81x
2
3. 16c
2
– 64
4. 27 – 3h
2
5. 3c
4
– 81c
6. x
4
– x
7. 5x
5
– 5000x
2
8. y
3
+ 64
9. a
4
b – ab
4
10. 4x
2
–16y
2
bonus. x
12
– y
12
