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.

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, ....

A number with 3 or more factors Not Prime

Ex: 8, 100, 36, 49820, etc.

Writing a number as a product of its prime factors

Ex: 20 = 2 2 5 = 2∙ ∙ 2 5∙

Composite Numbers:

Prime Factorization:

Write the prime factorization of 3300.3300

2 1650

2 825

5 165

5 33= 2 x 2 x 5 x 5 x 3 x 11

Prime #s – 2, 3, 5, 7, 11, 13, 17, 19,

23, 29, 31

3 11= 22 x 3 x 52 x 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

1981, 138 2, 69, 3, 46, 6, 23,

1, 198 2, 99, 3, 66, 6, 33, 11,18, 9, 22,

138

2 69

3 23

GCF = 2 x 3

Determine the greatest common factor of 138 and 198.

198

2 99

11

9

3 3= 6

Prime #s – 2, 3, 5, 7, 11, 13, 17, 19,

23, 29, 31

Multiply all common prime factors to find GCF

Determine the least common multiple of 18, 20, and 30.

18

2018, 36, 54, 72, 90, 108, 126, 144, 162, 180, 198

20, 40, 60, 80, 100, 120, 140, 160, 180, 200, 220

3030, 60, 90, 120, 150, 180, 210, 240, 270, 300, 330

Determine the least common multiple of 18, 20, and 30.

LCM = 22x32x5

18

2 9

30

2 15

Pick out the highest power of each prime factor, then multiply these to

find LCM= 4x9x5

20

2 10

3 3 2 5 3 5

= 2 x 32

= 22 x 5

= 2 x 3 x 5

=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.

x

x

40

16 16 16 16

40

Need to find when the multiples of each side are the same (LCM)

16

2 8

40

2 20

2 4 2 10

= 24

= 23x 5

2 2 2 5

LCM = 24 x 5LCM = 80The smallest square that could be tiled is 80cm by

80cm

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.

16

40

xx x

x

The length of the square must be a factor of the length of each side of the rectangle. We need to find

the GCF

16

2 8

40

2 20

2 4 2 10

2 2 2 5

GCF = 23 GCF = 8

The largest square that could be used to tile is 8cm by 8cm

x

§ 3.2Perfect Squares, Perfect Cubes, and

Their Roots

Determine the square root of 12961296

2 648

2 324

2 162

2 81

Prime #s – 2, 3, 5, 7, 11, 13, 17, 19,

23, 29, 31

9 9

= ∙

3 3 3 3

2∙

2∙

2∙

2∙

3∙

3∙

3 3

= ( )( )

= 36 ∙ 36

The square root of 1296 is 36

Determine the square root of 1296(check with a calculator)

1296 ) = ( 1296 ) = ( 1296 ) = (

1296 ) = ( 36

Determine the cube root of 17281728

2 864

2 432

2 216

2 108

Prime #s – 2, 3, 5, 7, 11, 13, 17, 19,

23, 29, 31

2 54

= ∙ ∙

2 27

3 9

2∙

2∙

2∙

2∙

2∙

2∙

3 3

= ( )( )( )

= 12 ∙ 12 ∙ 12The cube root of 1728 is 12 3 3

3

Determine the cube root of 1728(check with a calculator)

1728 ) = ( 1728 ) = ( 1728 ) = (

1728 ) = ( 12

3 3 3

3

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

4913 = x3√3 √317 = x

17 in

17 in

17 in

Area = length width∙

Side area = 17 17 = 17∙ 2

= 289 in2

SA = 6(289) =1734 in2

§ 3.3Common Factors of a Polynomial

The 3 types of factoring we will be learning:1. Greatest Common Factor2. Trinomial3. Difference of Squares

Example 1: Factor each binomial a) 6n + 9 b) 6c + 4c2

What is the GCF of 6 and 9?

= 3(

What is left over?

2n + 3)

What is the GCF of 6c and 4c2?

= 2c(

What is left over?

3 + 2c)

6n =9 =

6c =4c2 =

2 3∙ ∙n3∙3

2 3∙∙c2 2∙ ∙c∙c

Example 2: Factor each binomial a) 5 – 10z – 5z2 b) -12x3y – 20xy2 – 16x2y2

What is the GCF?

= 5(1

What is left over?

– 2z

What is the GCF?

= -4xy(3x2

What is left over?

+ 5y

5 =-10z = -12x3y =

– 20xy2 =

-2 5∙ ∙z

-5z2 = -5∙z∙z

– z2)

– 2 2 3∙ ∙ ∙x∙x∙x∙y – 2 2 5∙ ∙ ∙x∙y∙y

+ 4xy)

– 16x2y2 = – 2 2 2 2∙ ∙ ∙ ∙x x∙ ∙ y∙y

5

§ 3.5Polynomials of the Form x2 + bx + c

Expand the Brackets

(x – 4)(x + 2)

x +2

x x2 +2x

= x2 + 2x – 4x – 8

= x2 – 2x – 8

-4 -4x -8

Expand the Brackets

(8 – b)(3 – b)

3 -b

8 24 -8b

= 24 – 8b – 3b + b2= 24 – 11b + b2

-b -3b +b2

= b2 – 11b + 24

The 3 types of factoring we will be learning:1. Greatest Common Factor2. Trinomial3. Difference of Squares

Example 2: Factor each trinomial

a) 822 xx

= (x – 4)(x + 2)

No

-8

Is there a common factor?Step 1

Put first and last in box

2

3

4 Factor out GCF for each vertical and horizontal set.

(take sign of closest term)

-8

Multiply (+1)(-8)Look for numbers: ___ x ___ = -8 ___ + ___ = -2

+2

-4

Example 2: Factor each trinomial

b) 35122 aa No+35

Is there a common factor?Step 1

Put first and last in box

2

3

4 Factor out GCF for each vertical and horizontal set.

(take sign of closest term)

+35Multiply (+1)(+35)Look for numbers: ___ x ___ = +35 ___ + ___ = -12

-7

-5

Example 2: Factor each trinomial

c) No

-24

Is there a common factor?Step 1

Put first and last in box

2

3

4 Factor out GCF for each vertical and horizontal set.

(take sign of closest term)

-24

Multiply (+1)(-24)Look for numbers: ___ x ___ = -24 ___ + ___ = -5

-8

+3

2524 dd

2 5 24d d

Example 2: Factor each trinomial

d)

-32

Is there a common factor?Step 1

Put first and last in box

2

3

4 Factor out GCF for each vertical and horizontal set.

(take sign of closest term)

-32

Multiply (+1)(-32)Look for numbers: ___ x ___ = -32 ___ + ___ = +4

-4

+8

128164 2 tt Yes

= -4( t2 + 4t – 32)

§ 3.6Polynomials of the Form ax2 + bx + c

Expand and Simplify

+2

+4 +8

(3 4)(4 2)d d

Expand and Simplify

(-2g + 8)(7 – 3g)

+7 -3g

-2g -14g

+6g2

=-14g + 6g2 + 56 – 24g= 6g2 – 38g + 56

+8 +56

-24g

The 3 types of factoring we will be learning:

Example 2: Factor each trinomial

a)

1. Greatest Common Factor2. Trinomial3. Difference of Squares

9204 2 xx+36

Is there a common factor?Step 1

Put first and last in box

2

3

4 Factor out GCF for each vertical and horizontal set.

(take sign of closest term)

+9

Multiply (+4)(+9)Look for numbers: ___ x ___ = +36 ___ + ___ = +20

+9

+1

No

Example 2: Factor each trinomial

b) No

-210Is there a common factor?Step 1

Put first and last in box

2

3

4 Factor out GCF for each vertical and horizontal set.

(take sign of closest term)

-35Multiply (+6)(-35)Look for numbers: ___ x ___ = -210 ___ + ___ = -11

-7

+5

35116 2 aa

Example 2: Factor each trinomial

c) No

-30Is there a common factor?Step 1

Put first and last in box

2

3

4 Factor out GCF for each vertical and horizontal set.

(take sign of closest term)

-10Multiply (+3)(-10)Look for numbers: ___ x ___ = -30 ___ + ___ = -13

-5

+2

10133 2 dd

Example 2: Factor each trinomial

d)+6

Is there a common factor?Step 1

Put first and last in box

2

3

4 Factor out GCF for each vertical and horizontal set.

(take sign of closest term)

+3

Multiply (+2)(+3)Look for numbers: ___ x ___ = +6 ___ + ___ = -7

-3

-1

Yes

= 3(2t2 – 7t + 3)

3

9216 2 tt

§3.7Multiplying Polynomials

+5

)43(52 2 hhh

Example #1: Expand and Simplify

a)

-4

-20

Example #1: Expand and Simplify

b)

-6)64)(233( 22 ffff

-2 +12

Example #1: Expand and Simplify

c) 252 tr )52)(52( trtr

Example #1: Expand and Simplify

d)

+5

-10y

)534)(23( yxyx

Example #1: Expand and Simplify

e) )13)(3(3)5)(32( cccc

Example #1: Expand and Simplify

f) 223)42)(13( yxxyx 3 2 3 2x y x y

§3.8Factoring Special Polynomials

Example 1: Factor the following

a) No

+36Is there a common factor?Step 1

Put first and last in box

2

3

4 Factor out GCF for each vertical and horizontal set.

(take sign of closest term)

+9Multiply (+4)(+9)Look for numbers: ___ x ___ = +36 ___ + ___ = +12

+3

+3

9124 2 xx

Example 1: Factor the following

b) No

+100

Is there a common factor?Step 1

Put first and last in box

2

3

4 Factor out GCF for each vertical and horizontal set.

(take sign of closest term)

+4

Multiply (+25)(+4)Look for numbers:

___ x ___ = +100 ___ + ___ = -20

-2

-2

225 20 4x x

225204 xx

Example 1: Factor the following

c) No

+6Is there a common factor?Step 1

Put first and last in box

2

3

4 Factor out GCF for each vertical and horizontal set.

(take sign of closest term)

Multiply (+2)(+3)Look for numbers: ___ x ___ = +6 ___ + ___ = -7

22 372 yxyx

= 2x2 – 6xy – xy + 3y2

= 2x2 – 7xy + 3y2

Check Answer:

Example 1: Factor the following

d) No

-20Is there a common factor?Step 1

Put first and last in box

2

3

4 Factor out GCF for each vertical and horizontal set.

(take sign of closest term)

Multiply (+10)(-2)Look for numbers: ___ x ___ = -20 ___ + ___ = -1

22 210 fdfd

= 10d2 – 5df + 4df – 2f 2

= 10d2 – df – 2f 2

Check Answer:

Example 2: Factor each Difference of Squares

a) No

-900

Is there a common factor?Step 1

Put first and last in box

2

3

4 Factor out GCF for each vertical and horizontal set.

(take sign of closest term)

Multiply (+25)(-36)Look for numbers:

___ x ___ = -900

___ + ___ = 0

23625 x

225 0 36x x

25

236x

-30 +30 -30 +30put in box with x

30x

30x

5 6x5

6x

(5 6 )(5 6 )x x

Example 2: Factor each Difference of Squares

b) No

-3969

Is there a common factor?Step 1

Put first and last in box

2

3

4 Factor out GCF for each vertical and horizontal set.

(take sign of closest term)

Multiply (81)(-49)Look for numbers:

___ x ___ = -3969

___ + ___ = 0

281 0 49m m

281m

49

-63 +63 -63 +63put in box with m

63m

63m

9m 79m

7

(9 7)(9 7)m m

4981 2 m

Example 2: Factor each difference of squares

c)

Three rules to be a difference of squaresAll of the exponents must be _______________.All of the coefficients (numbers) must be ____________.The two terms must be joined with a _______________.

Squares (Even)Square Numbers

Subtraction Sign

= 5(x2 4y2)(x2 4y2)+ –

Is there a common factor?Step 1

Take the square root of both terms and separate into two sets of brackets.

2

One Positive and One Negative

3

Yes44 805 yx

Difference of Squares

= 5(x4 – 16y4)

= 5(x2 + 4y2) (x 2y)(x 2y)+ –

Example 2: Factor each difference of squares

d)

= 2(9v2 w2)(9v2 w2)+ –

Is there a common factor?Step 1

Take the square root of both terms and separate into two sets of brackets.

2

One Positive and One Negative

3

Yes

44 2162 wv Difference of Squares

= 2(81v4 – w4)

= 2(9v2 + w2)(3v w)(3v w)+ –