Chapter 3 Factors & Products

3.1 – Factors & Multiples of Whole 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, .... Prime Numbers: 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

3300 Write the prime factorization of 3300. 2 1650 2 825 5 165 Prime #s – 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31 2 1650 2 825 5 165 = 2 x 2 x 5 x 5 x 3 x 11 5 33 = 22 x 3 x 52 x 11 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 The least multiple that is a multiple of each number in a set.  LCM Ex: 84 is the LCM of 12 and 21 Least Common Multiple:

Determine the greatest common factor of 138 and 198. 1, 138 2, 69, 3, 46, 6, 23, 198 1, 198 2, 99, 3, 66, 6, 33, 9, 22, 11,18,

Multiply all common prime factors to find GCF Determine the greatest common factor of 138 and 198. Prime #s – 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31 198 138 2 99 2 69 11 9 3 23 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. 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. 9 2 10 2 15 3 3 5 3 2 5 = 22 x 5 = 2 x 3 x 5 = 2 x 32 Pick out the highest power of each prime factor, then multiply these to find LCM LCM = 22x32x5 = 4x9x5 =180

Need to find when the multiples of each side are the same (LCM) 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. 16 40 Need to find when the multiples of each side are the same (LCM) 2 8 2 20 2 40 4 2 10 x 2 2 5 2 40 = 24 = 23x 5 LCM = 24 x 5 16 16 16 16 x LCM = 80 The 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 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 8 2 20 2 4 2 10 40 x 2 2 5 2 x GCF = 23 x GCF = 8 x x 16 The largest square that could be used to tile is 8cm by 8cm

Perfect Squares, Perfect Cubes, and Their Roots § 3.2 Perfect Squares, Perfect Cubes, and Their Roots

1296 Determine the square root of 1296 2 648 2 324 = ∙ 2∙ 2∙ 2∙ 2∙ 3∙ Prime #s – 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31 2 648 2 324 = ∙ 2∙ 2∙ 2∙ 2∙ 3∙ 3∙ 3 3 2 162 = ( )( ) 2 81 = 36 ∙ 36 9 9 The square root of 1296 is 36 3 3 3 3

Determine the square root of 1296 (check with a calculator) 1296 ) = 36 ( 1296 ) = ( 1296 ) = ( 1296 ) =

1728 Determine the cube root of 1728 864 2 432 2 216 2 = ∙ ∙ 2∙ 2∙ 2∙ Prime #s – 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31 864 2 432 2 216 2 = ∙ ∙ 2∙ 2∙ 2∙ 2∙ 2∙ 2∙ 3 3 3 108 2 = ( )( )( ) 54 2 = 12 ∙ 12 ∙ 12 2 27 The cube root of 1728 is 12 3 9 3 3

Determine the cube root of 1728 (check with a calculator) 3 ( 1728 ) = 12 3 ( 1728 ) = 3 ( 1728 ) = ( 1728 ) = 3

A cube has volume 4913 cubic inches A cube has volume 4913 cubic inches. What is the surface area of the cube? Volume = length ∙ width ∙ height = x ∙ x ∙ x = x3 Volume = 4913 in3 √ 3 √ 3 4913 = x3 17 in 17 = x Area = length ∙ width Side area = 17 ∙ 17 = 172 = 289 in2 17 in 17 in SA = 6(289) = 1734 in2

Common Factors of a Polynomial § 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? What is the GCF of 6c and 4c2? 6n = 2∙3∙n 6c = 2∙3∙c What is left over? 9 = 3∙3 What is left over? 4c2 = 2∙2∙c∙c = 3( 2n + 3) = 2c( 3 + 2c)

5 = 5 -10z = -2∙5∙z -12x3y = – 2∙2∙3∙x∙x∙x∙y -5z2 = -5∙z∙z – 20xy2 = Example 2: Factor each binomial a) 5 – 10z – 5z2 b) -12x3y – 20xy2 – 16x2y2 What is the GCF? What is the GCF? 5 = 5 What is left over? -10z = -2∙5∙z -12x3y = – 2∙2∙3∙x∙x∙x∙y -5z2 = -5∙z∙z – 20xy2 = – 2∙2∙5∙x∙y∙y What is left over? – 16x2y2 = – 2∙2∙2∙2∙x∙x∙ y∙y = 5( 1 – 2z – z2) = -4xy( 3x2 + 5y + 4xy)

Polynomials of the Form x2 + bx + c § 3.5 Polynomials of the Form x2 + bx + c

(x – 4)(x + 2) x +2 x -4 Expand the Brackets = x2 – 2x – 8 -8

(8 – b)(3 – b) 3 -b 8 -b Expand the Brackets = b2 – 11b + 24

(take sign of closest term) The 3 types of factoring we will be learning: Is there a common factor? Step 1 1. Greatest Common Factor 2. Trinomial 3. Difference of Squares No Put first and last in box 2 Example 2: Factor each trinomial a) -8 3 Multiply (+1)(-8) Look for numbers: ___ x ___ = -8 ___ + ___ = -2     +2       -4   -8 4 Factor out GCF for each vertical and horizontal set. (take sign of closest term) = (x – 4)(x + 2)

(take sign of closest term) Example 2: Factor each trinomial b) +35 Is there a common factor? Step 1 No Put first and last in box 2   -7       3 Multiply (+1)(+35) -5   +35 Look for numbers: ___ x ___ = +35 ___ + ___ = -12     4 Factor out GCF for each vertical and horizontal set. (take sign of closest term)

(take sign of closest term) Example 2: Factor each trinomial c) Is there a common factor? Step 1 No   -24 Put first and last in box 2 3   Multiply (+1)(-24) -8 Look for numbers: ___ x ___ = -24 ___ + ___ = -5         +3   -24 4 Factor out GCF for each vertical and horizontal set. (take sign of closest term)  

(take sign of closest term) Example 2: Factor each trinomial d) Is there a common factor? Step 1 Yes -32 Put first and last in box 2 = -4( t2 + 4t – 32) 3 Multiply (+1)(-32)   -4 Look for numbers: ___ x ___ = -32 ___ + ___ = +4         +8   -32 4 Factor out GCF for each vertical and horizontal set. (take sign of closest term)    

Polynomials of the Form ax2 + bx + c § 3.6 Polynomials of the Form ax2 + bx + c

Expand and Simplify   +2           +4   +8

-3g -2g +8 (-2g + 8)(7 – 3g) +7 = 6g2 – 38g + 56 Expand and Simplify +56 -24g

(take sign of closest term) The 3 types of factoring we will be learning: 1. Greatest Common Factor 2. Trinomial 3. Difference of Squares Is there a common factor? Step 1 No Put first and last in box 2 Example 2: Factor each trinomial a) +36 3 Multiply (+4)(+9) Look for numbers: ___ x ___ = +36 ___ + ___ = +20   +9         +1   +9 4 Factor out GCF for each vertical and horizontal set. (take sign of closest term)  

(take sign of closest term) Example 2: Factor each trinomial b) -210 Is there a common factor? Step 1 No   -7 Put first and last in box 2       3 Multiply (+6)(-35) +5   -35 Look for numbers: ___ x ___ = -210 ___ + ___ = -11     4 Factor out GCF for each vertical and horizontal set. (take sign of closest term)

(take sign of closest term) Example 2: Factor each trinomial c) -30 Is there a common factor? Step 1 No   -5 Put first and last in box 2       3 Multiply (+3)(-10) +2   -10 Look for numbers: ___ x ___ = -30 ___ + ___ = -13     4 Factor out GCF for each vertical and horizontal set. (take sign of closest term)

(take sign of closest term) Example 2: Factor each trinomial d) Is there a common factor? Step 1 Yes +6 Put first and last in box 2 = 3( 2t2 – 7t + 3)   -3 3 Multiply (+2)(+3) Look for numbers: ___ x ___ = +6 ___ + ___ = -7         -1   +3 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 -4             +5     -20          

Example #1: Expand and Simplify b)     -6                 -2     +12            

Example #1: Expand and Simplify c)                        

Example #1: Expand and Simplify +5                   -10y            

Example #1: Expand and Simplify e)                                                                

Example #1: Expand and Simplify f)                                                                      

§3.8 Factoring Special Polynomials

(take sign of closest term) Example 1: Factor the following a) +36 Is there a common factor? Step 1 No   +3 Put first and last in box 2       3 Multiply (+4)(+9) +3   +9 Look for numbers: ___ x ___ = +36 ___ + ___ = +12     4 Factor out GCF for each vertical and horizontal set. (take sign of closest term)  

(take sign of closest term) Example 1: Factor the following b) Is there a common factor? Step 1 No   +100 Put first and last in box 2 3 Multiply (+25)(+4)   -2 Look for numbers: ___ x ___ = +100 ___ + ___ = -20         -2   +4 4 Factor out GCF for each vertical and horizontal set. (take sign of closest term)    

(take sign of closest term) Example 1: Factor the following c) +6 Is there a common factor? Step 1 No     Put first and last in box 2       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) Check Answer: = 2x2 – 6xy – xy + 3y2 = 2x2 – 7xy + 3y2

(take sign of closest term) Example 1: Factor the following d) -20 Is there a common factor? Step 1 No     Put first and last in box 2       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) Check Answer: = 10d2 – 5df + 4df – 2f 2 = 10d2 – df – 2f 2

(take sign of closest term) Example 2: Factor each Difference of Squares a) Is there a common factor? Step 1 No -900 Put first and last in box 2 3 Multiply (+25)(-36) Look for numbers: ___ x ___ = -900 ___ + ___ = 0 -30 +30 -30 +30 put in box with 4 Factor out GCF for each vertical and horizontal set. (take sign of closest term)

(take sign of closest term) Example 2: Factor each Difference of Squares b) Is there a common factor? Step 1 No -3969 Put first and last in box 2 3 Multiply (81)(-49) Look for numbers: ___ x ___ = -3969 ___ + ___ = 0 -63 +63 -63 +63 put in box with 4 Factor out GCF for each vertical and horizontal set. (take sign of closest term)

= 5(x4 – 16y4) = 5(x2 4y2)(x2 4y2) + – = 5(x2 + 4y2) (x 2y)(x 2y) + – Three rules to be a difference of squares All 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 Example 2: Factor each difference of squares c) Is there a common factor? Step 1 Yes = 5(x4 – 16y4) Take the square root of both terms and separate into two sets of brackets. 2 = 5(x2 4y2)(x2 4y2) + – = 5(x2 + 4y2) (x 2y)(x 2y) + – One Positive and One Negative 3 Difference of Squares

= 2(81v4 – w4) = 2(9v2 w2)(9v2 w2) + – = 2(9v2 + w2) (3v w)(3v w) + – Example 2: Factor each difference of squares d) Difference of Squares Is there a common factor? Step 1 Yes = 2(81v4 – w4) = 2(9v2 w2)(9v2 w2) + – Take the square root of both terms and separate into two sets of brackets. 2 = 2(9v2 + w2) (3v w)(3v w) + – One Positive and One Negative 3