Download +2 - Sardis Secondary

Chapter 3
Factors & Products
3.1 – Factors & Multiples of Whole
Numbers
Prime Numbers: A whole number with exactly 2
factors  1 and the number itself
Ex: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, ....
Composite Numbers: A number with 3 or more
factors Not Prime
Ex: 8, 100, 36, 49820, etc.
Prime Factorization: Writing a number as a product
of its prime factors
Ex: 20 = 2∙2∙5 = 22∙5
Write the prime factorization of 3300.
3300
2
Prime #s – 2, 3, 5,
7, 11, 13, 17, 19,
23, 29, 31
1650
2
825
5
= 2 x 2 x 5 x 5 x 3 x 11
= 22 x 3 x 52 x 11
165
5
33
3
11
Greatest Common Factor: The greatest number
that divides into each number in a set  GCF
Ex: 5 is the GCF of 10 and 15
Least Common Multiple: The least multiple that is a
multiple of each number in a set.  LCM
Ex: 84 is the LCM of 12 and 21
Determine the greatest common factor of 138 and 198.
138
1, 2, 3, 6, 23, 46, 69, 138
198
1, 2, 3, 6, 9, 11,18, 22,33, 66, 99, 198
Determine the greatest common factor of 138 and 198.
138
2
Prime #s – 2, 3, 5,
7, 11, 13, 17, 19,
23, 29, 31
99
2
69
3
198
23
11
9
3
3
GCF = 2 x 3 = 6
Multiply all common prime factors to find GCF
Determine the least common multiple of 18, 20, and 30.
18
18, 36, 54, 72, 90, 108, 126, 144, 162, 180, 198
20
20, 40, 60, 80, 100, 120, 140, 160, 180, 200, 220
30
30, 60, 90, 120, 150, 180, 210, 240, 270, 300, 330
Determine the least common multiple of 18, 20, and 30.
2
30
20
18
3
3
= 2 x 32
10
2
9
2
= 22 x 5
15
2
3
5
5
=2x3x5
Pick out the highest power of each prime
factor, then multiply these to find LCM
LCM = 22x32x5 = 4x9x5 =180
What is the side length of the smallest square that could be tiled
with rectangles that measure 16cm by 40cm? Assume the
rectangles cannot be cut. Sketch the square and rectangles.
40 the
16 Need to find when
multiples of each side are
the same (LCM)
2
x
2
40
= 24
16 16 16 16
x
2
4
2
40
20
2
8
10
2
2
= 23x 5
LCM = 24 x 5
LCM = 80
The smallest square that could be tiled is 80cm by 80cm
5
What is the side length of the largest square that could be used to
tile a rectangle that measures 16cm by 14cm? Assume that the
squares cannot be cut. Sketch the rectangle and squares.
40
16
The length of the square must be a
factor of the length of each side of the
rectangle. We need to find the GCF
2
2
40 x
x
x
x
2
4
2
x
20
2
8
2
10
2
5
GCF = 23
GCF = 8
16
The largest square that could be used to tile is 8cm by 8cm
§ 3.2
Perfect Squares, Perfect Cubes, and
Their Roots
Determine the square root of 1296
1296
2
648
2
= 2∙ 2∙ 2∙ 2∙3∙ 3∙ 3 ∙3
=(
)(
Prime #s – 2, 3, 5,
7, 11, 13, 17, 19,
23, 29, 31
)
324
2
162
2
= 36 ∙ 36
The square root of
1296 is 36
81
9
9
3
3
3
3
Determine the square root of 1296
( 1296 ) =
(check with a calculator)
( 1296 ) =
( 1296 ) =
36
( 1296 ) =
Determine the cube root of 1728
1728
864
2
432
2
= 2∙ 2∙ 2∙ 2∙2∙ 2∙ 3 ∙3 ∙3
=(
)(
)(
Prime #s – 2, 3, 5,
7, 11, 13, 17, 19,
23, 29, 31
216
2
)
= 12 ∙ 12 ∙ 12
The cube root of 1728 is 12
108
2
54
2
27
2
3
9
3
3
Determine the cube root of 1728
3
(check with a calculator)
3
( 1728 ) =
3
( 1728 ) =
( 1728 ) = 12
3
( 1728 ) =
A cube has volume 4913 cubic inches. What
is the surface area of the cube?
Volume = 4913 in3
Volume = length ∙ width ∙ height
=x∙x∙x
= x3
3
3
√4913 =√x
17 in
Area =
length ∙ width
3
17 = x
Side area = 17 ∙ 17 = 172
= 289 in2
17 in
17 in
SA = 6(289) = 1734 in2
§ 3.3
Common Factors of a Polynomial
The 3 types of factoring we will be learning:
1. Greatest Common Factor
2. Trinomial
3. Difference of Squares
Example 1: Factor each binomial
a) 6n + 9
b) 6c + 4c2
What is the GCF
of 6 and 9?
6n = 2∙3∙n
9 = 3∙3
What is left over?
= 3( 2n + 3)
What is the GCF
of 6c and 4c2?
6c = 2∙3∙c
4c2 = 2∙2∙c∙c
= 2c(3 + 2c)
What is left over?
Example 2: Factor each binomial
a) 5 – 10z – 5z2
What is the GCF?
5=5
-10z = -2∙5∙z
-5z2 = -5∙z∙z
What is left over?
= 5( 1 – 2z – z2)
b) -12x3y – 20xy2 – 16x2y2
What is the GCF?
What is left over?
-12x3y = – 2∙2∙3∙x∙x∙x∙y
– 20xy2 = – 2∙2∙5∙x∙y∙y
– 16x2y2 = – 2∙2∙2∙2∙x∙x∙ y∙y
= -4xy(3x2 + 5y + 4xy)
§ 3.5
Polynomials of the Form x2 + bx + c
Expand the Brackets
(x – 4)(x + 2)
x
+2
x
x2
+2x
-4
-4x
-8
= x2 + 2x – 4x – 8
= x2 – 2x – 8
Expand the Brackets
(8 – b)(3 – b)
3
-b
8
24
-8b
-b
-3b
+b2
= 24 – 8b – 3b + b2
= 24 – 11b + b2
= b2 – 11b + 24
The 3 types of factoring we will be learning:
1. Greatest Common Factor
2. Trinomial
3. Difference of Squares
Step 1
2
Example 2: Factor each trinomial
-8
a) x 2  2 x  8
-8
= (x – 4)(x + 2)
Put first and last
in box
3
Multiply (+1)(-8)
Look for numbers:
___ x ___ = -8
___ + ___ = -2
4
Factor out GCF for
each vertical and
horizontal set.
(take sign of
closest term)
+2
-4
Is there a common
factor? No
Example 2: Factor each trinomial
+35
2
b) a  12a  35
-7
-5
Step 1
2
Is there a common
factor? No
Put first and last
in box
3
Multiply (+1)(+35)
Look for numbers:
___ x ___ = +35
___ + ___ = -12
4
Factor out GCF for
each vertical and
horizontal set.
(take sign of
closest term)
+35
Example 2: Factor each trinomial
c)
 24  5d  d
2
Step 1
2
-24
d 2  5d  24
-8
+3
Is there a common
factor? No
Put first and last
in box
3
Multiply (+1)(-24)
Look for numbers:
___ x ___ = -24
___ + ___ = -5
4
Factor out GCF for
each vertical and
horizontal set.
(take sign of
closest term)
-24
Example 2: Factor each trinomial
d)
 4t 2  16t  128
-32
Step 1
2
= -4( t2 + 4t – 32)
-32
Put first and last
in box
3
Multiply (+1)(-32)
Look for numbers:
___ x ___ = -32
___ + ___ = +4
4
Factor out GCF for
each vertical and
horizontal set.
(take sign of
closest term)
-4
+8
Is there a common
factor? Yes
§ 3.6
Polynomials of the Form ax2 + bx + c
Expand and Simplify
(3d  4)(4d  2)
+2
+4
+8
Expand and Simplify
(-2g + 8)(7 – 3g)
+7
-3g
-2g
-14g
+6g2
+8
+56
-24g
=-14g + 6g2 + 56 – 24g
= 6g2 – 38g + 56
The 3 types of factoring we will be learning:
1. Greatest Common Factor
2. Trinomial
3. Difference of Squares
Step 1
2
Example 2: Factor each trinomial
+36
a)
2
4 x  20 x  9
+1
3
+9
+9
4
Is there a common
factor? No
Put first and last
in box
Multiply (+4)(+9)
Look for numbers:
___ x ___ = +36
___ + ___ = +20
Factor out GCF for
each vertical and
horizontal set.
(take sign of
closest term)
Example 2: Factor each trinomial
-210
b)
6a  11a  35
2
Step 1
-7
+5
-35
2
3
4
Is there a common
factor? No
Put first and last
in box
Multiply (+6)(-35)
Look for numbers:
___ x ___ = -210
___ + ___ = -11
Factor out GCF for
each vertical and
horizontal set.
(take sign of
closest term)
Example 2: Factor each trinomial
-30
c)
3d  13d  10
2
-5
+2
-10
Step 1
2
Is there a common
factor? No
Put first and last
in box
3
Multiply (+3)(-10)
Look for numbers:
___ x ___ = -30
___ + ___ = -13
4
Factor out GCF for
each vertical and
horizontal set.
(take sign of
closest term)
Example 2: Factor each trinomial
d)
6t  21t  9
2
Step 1
+6
=
3(2t2
– 7t + 3)
-3
-1
3
2
Is there a common
factor? Yes
Put first and last
in box
3
Multiply (+2)(+3)
Look for numbers:
___ x ___ = +6
___ + ___ = -7
4
Factor out GCF for
each vertical and
horizontal set.
(take sign of
closest term)
+3
§3.7
Multiplying Polynomials
Example #1: Expand and Simplify
a)
2h  5(h 2  3h  4)
-4
+5
-20
Example #1: Expand and Simplify
b)
(3 f 2  3 f  2)(4 f 2  f  6)
-6
-2
+12
Example #1: Expand and Simplify
c)
2r  5t   (2r  5t )( 2r  5t )
2
Example #1: Expand and Simplify
d)
(3x  2 y )( 4 x  3 y  5)
+5
-10y
Example #1: Expand and Simplify
e)
(2c  3)(c  5)  3(c  3)( 3c  1)
Example #1: Expand and Simplify
  3x  2 y 3x  2 y 
2
f) (3x  y  1)( 2 x  4)  3x  2 y 
§3.8
Factoring Special Polynomials
Example 1: Factor the following
+36
a)
4 x 2  12 x  9
Step 1
+3
+3
+9
2
3
4
Is there a common
factor? No
Put first and last
in box
Multiply (+4)(+9)
Look for numbers:
___ x ___ = +36
___ + ___ = +12
Factor out GCF for
each vertical and
horizontal set.
(take sign of
closest term)
Example 1: Factor the following
b)
4  20 x  25 x
2
Step 1
2
+100
25 x 2  20 x  4
-2
-2
Is there a common
factor? No
Put first and last
in box
3
Multiply (+25)(+4)
Look for numbers:
___ x ___ = +100
___ + ___ = -20
4
Factor out GCF for
each vertical and
horizontal set.
(take sign of
closest term)
+4
Example 1: Factor the following
+6
c)
2 x  7 xy  3 y
2
2
Step 1
2
Check Answer:
= 2x2 – 6xy – xy + 3y2
= 2x2 – 7xy + 3y2
Is there a common
factor? No
Put first and last
in box
3
Multiply (+2)(+3)
Look for numbers:
___ x ___ = +6
___ + ___ = -7
4
Factor out GCF for
each vertical and
horizontal set.
(take sign of
closest term)
Example 1: Factor the following
-20
d) 10d 2
 df  2 f 2
Step 1
2
Check Answer:
= 10d2 – 5df + 4df – 2f 2
= 10d2 – df – 2f 2
Is there a common
factor? No
Put first and last
in box
3
Multiply (+10)(-2)
Look for numbers:
___ x ___ = -20
___ + ___ = -1
4
Factor out GCF for
each vertical and
horizontal set.
(take sign of
closest term)
Example 2: Factor each Difference of Squares
a)
25  36 x 2
Step 1
-900
25  0 x  36 x
5 6x
5 25 30x
2
2
6x 30x
3
Put first and last
in box
Multiply (+25)(-36)
Look for numbers:
-30
___ x +30
___ = -900
-30
___ + +30
___ = 0
put in box with x
36x 2
 (5  6 x)(5  6 x)
Is there a common
factor? No
4
Factor out GCF for
each vertical and
horizontal set.
(take sign of
closest term)
Example 2: Factor each Difference of Squares
b)
81m  49
Step 1
-3969
2
2
81m  0m  49
9m 7
2
63m
81m
9m
2
7
3
Put first and last
in box
Multiply (81)(-49)
Look for numbers:
-63
___ x +63
___ = -3969
-63
___ + +63
___ = 0
put in box with m
63m 49
 (9m  7)(9m  7)
Is there a common
factor? No
4
Factor out GCF for
each vertical and
horizontal set.
(take sign of
closest term)
Three rules to be a difference of squares
Squares (Even)
All of the exponents must be _______________.
Square Numbers
All of the coefficients (numbers) must be ____________.
Subtraction Sign
The two terms must be joined with a _______________.
Example 2: Factor each difference of squares
c)
5 x 4  80 y 4
Step 1
= 5(x4 – 16y4)
2
– 2)
= 5(x2 +4y2)(x2 4y
= 5(x2 + 4y2) (x +2y)(x – 2y)
Difference of Squares
3
Is there a common
factor? Yes
Take the square root
of both terms and
separate into two sets
of brackets.
One Positive and
One Negative
Example 2: Factor each difference of squares
d)
162v  2 w
4
4
Difference of Squares
= 2(81v4 – w4)
Step 1
= 2(9v2 + w2)(9v2 – w2)
2
= 2(9v2 + w2)(3v + w)(3v – w)
3
Is there a common
factor? Yes
Take the square root
of both terms and
separate into two sets
of brackets.
One Positive and
One Negative