10/16/2015Math KM1 Chapter 6: Introduction to Polynomials and Polynomial Functions 6.1 Introduction to Factoring 6.2 FactoringTrinomials: x 2 +bx+c 6.3 FactoringTrinomials: ax 2 +bx+c 6.4 Special Factoring 6.5 Factoring: A General Strategy 6.6 Applications

10/16/2015Math KM2 6.1 Introduction to Factoring 6.1

10/16/2015Math KM3 Let’s Build the Greatest Common Factor of 90x 2 y 3 z and 50y 4 z 5 The GCF of 90x 2 y 3 z and 50y 4 z 5 is the product of the “common” bases raised to the smallest exponent. or 6.1

10/16/2015Math KM4 Let’s Build the Greatest Common Factor of 21x 2 z and 10y 4 21x 2 z and 10y 4 have no common factors! The only factor common to both expressions is 1. 21x 2 z and 10y 4 are RELATIVELY PRIME because their GCF is

10/16/2015Math KM5 Factoring out the GCF is Reversing the Distributive Property 6.1

10/16/2015Math KM6 Factor out the GCF from 12x x 3 12x 5 20x 3 4x 3 3x 2 5 4x 3 3x

10/16/2015Math KM7 Factor: 12x x 3 12x x 3 = 4x 3 (3x 2 + 5) 6.1

10/16/2015Math KM8 Factor: 6x 2 y 5 - 8x 3 y 4 6x 2 y 5 - 8x 3 y 4 = 2x 2 y 4 (3y - 4x) 6.1

10/16/2015Math KM9 Factor: 9x 3 – 11y x 3 – 11y

10/16/2015Math KM10 Factoring 6.1

10/16/2015Math KM11 Factor a Tricky One! x(x + 2) – 6(x + 2) = ( x + 2 )( x – 6 ) 6.1

10/16/2015Math KM12 Another Tricky One! (x - 7)3x +(x - 7)5 = ( x - 7 )( 3x + 5 ) 6.1

10/16/2015Math KM13 Factor by Grouping: Example 1: REVERSE FOIL ab + 7b – 3a – 21 = b(a + 7)– 3(a + 7) = (a + 7)(b - 3) (a + 7)(b – 3) = ab – 3a + 7b

10/16/2015Math KM14 Factor by Grouping: Example 2: REVERSE FOIL x 2 + 3x + 5x + 15 = x(x + 3) + 5(x + 3) = (x + 3)(x + 5) 6.1

10/16/2015Math KM15 Factor by Grouping: Example 3: REVERSE FOIL x 2 + 5x – 5x - 25 = x(x + 5) - 5(x + 5) = (x + 5)(x - 5) 6.1

10/16/2015Math KM16 Factor by Grouping: Example 4: REVERSE FOIL x 2 - 9x + 11x - 99 = x(x - 9) + 11(x - 9) = (x + 11)(x - 9) 6.1

10/16/2015Math KM17 Factor by Grouping: Example 5: REVERSE FOIL x x x = x 2 (x - 10) - 10(x - 10) = (x 2 – 10)(x - 10) 6.1

10/16/2015Math KM18 Factor by Grouping: Example 6: REVERSE FOIL 18x x + 30x - 35 = 3x(6x - 7) + 5(6x - 7) = (6x - 7)(3x + 5) 6.1

10/16/2015Math KM19 Factor by Grouping: Example 7: REVERSE FOIL 25x x + 35x + 49 = 5x(5x + 7)+ 5(5x + 7) = (5x + 7) (5x + 7)

10/16/2015Math KM20 Where We Left Off Last Class

10/16/2015Math KM & 4.5 FactoringTrinomials: ax 2 +bx+c 6.2

10/16/2015Math KM22 First, Let’s Review Factor by Grouping ab + 7b – 3a – 21 x 2 + 2x + 10x

10/16/2015Math KM23 Now, Let’s Review FOIL! Aha! FL = OI (8)(-15) = (10)(-12) -120 =

10/16/2015Math KM24 What’s the Diamond? ax 2 + bx + c Add to b Multiply to ac 6.2

10/16/2015Math KM25 2x x -40 Add to -11 Multiply to

10/16/2015Math KM26 2x x -40 Add to -11 Multiply to

10/16/2015Math KM27 6x x +12 Add to -17 Multiply to

10/16/2015Math KM28 Start with the GCF 6.2

10/16/2015Math KM29 More Problems? 12y y 2 – 70y + 15x - 4x ax ax 2 – 160ax 2x 4 + 5x x 6 + 4x 3 –

10/16/2015Math KM Special Factoring 6.3

10/16/2015Math KM31 Special Factoring Shortcuts 6.3

10/16/2015Math KM32 Special Polynomials 6.3

10/16/2015Math KM33 Perfect Trinomial Square 6.3

10/16/2015Math KM34 Perfect Trinomial Square 6.3

10/16/2015Math KM35 Perfect Trinomial Square 6.3

10/16/2015Math KM36 Perfect Trinomial Square 6.3

10/16/2015Math KM37 OK – Short Cut Time! 6.3

10/16/2015Math KM38 Difference of Squares 6.3

10/16/2015Math KM39 You can do this! 6.3

10/16/2015Math KM40 Check these out! 6.3

10/16/2015Math KM41 Sum or Difference of Cubes nn cubed …… nn3n3 6.3

10/16/2015Math KM42 Sum or Difference of Cubes 6.3

10/16/2015Math KM43 Sum or Difference of Cubes 6.3

10/16/2015Math KM44 Sum or Difference of Cubes 6.3

10/16/2015Math KM45 How about a harder one? 6.3

10/16/2015Math KM Factoring: A General Strategy 6.4

10/16/2015Math KM47 Factoring Strategy GCF 1) GREATEST COMMON FACTOR Check carefully to see if there is a GCF and factor it out. If the leading coefficient is negative, factor out

10/16/2015Math KM48 Factoring Strategy Number of Terms 2) Number of TERMS a) Four Terms: Try grouping b) Three Terms: i) a 2 + 2ab + b 2 Perfect Square ii) a 2 – 2ab + b 2 Perfect Square iii) ax 2 + bx + c UNFOIL c) Two Terms: i) a 2 - b 2 Difference of Squares ii) a 2 + b 2 Sum of Squares - NF iii) x 3 – y 3 Difference of Cubes iv) x 3 + y 3 Sum of Cubes 6.4

10/16/2015Math KM49 Factor Completely: Example 1 2x 3 + 6x 2 – 8x - 24 = 2[ x 3 + 3x 2 – 4x – 12 ] = 2[ x 2 (x + 3) – 4(x+3) ] = 2[(x + 3)(x 2 – 4)] = 2[(x + 3)(x + 2)(x - 2)] = 2(x + 3)(x + 2)(x - 2) 6.4

10/16/2015Math KM50 Factor Completely: Example 2 5x x x = 5x[ x 2 – 16x + 64 ] = 5x[(x - 8)(x - 8)] = 5x(x - 8) 2 6.4

10/16/2015Math KM51 Factor Completely: Example 3 9x x = 9x 2 -3x + 15x - 5 = 3x(3x – 1) + 5(3x - 1) = (3x – 1)(3x + 5) 6.4

10/16/2015Math KM52 Factor Completely: Example 4 125x 3 + 8y 3 = (5x + 2y)(25x 2 – 10xy + 4y 2 ) = (5x) 3 + (2y) 3 6.4

10/16/2015Math KM53 Factor Completely: Example 5 x x – y = x x + 25 – y 2 = (x + 5) 2 – y 2 = [(x + 5) + y] [(x + 5) - y] = (x y)(x + 5 – y) 6.4

10/16/2015Math KM Applications 6.4

10/16/2015Math KM55 General Strategy for Solving Equations Using The Zero Factor Property 1) Arrange the equation so that one side is zero. 2) Completely factor the other side. 3) Set each factor equal to zero and solve, if possible. 4) Write the solution set. 5) Check each solution by substitution. 6.5

10/16/2015Math KM56 Zero Factor Property Solve: 2x(x + 5)(x-3) = 0 6.5

10/16/2015Math KM57 Zero Factor Property Solve: (2x - 7)(4x + 3)= 0 6.5

10/16/2015Math KM58 Zero Factor Property Solve: 6x 2 = 3x Use the properties of equality to rearrange the terms of the equation so that it is equal to ZERO. 6.5

10/16/2015Math KM59 Solve: x 2 = 169 or 6.5

10/16/2015Math KM60 Solve: x = 10x or 6.5

10/16/2015Math KM61 Solve: 3x 2 = 2 - x or 6.5

10/16/2015Math KM62 Solve: x + 12 = x(x – 3) or 6.5

10/16/2015Math KM63 Solve: 2x 3 + 3x 2 = 18x + 27 or 6.5

10/16/2015Math KM64 The Pool is Cool! Pat has a rectangular swimming pool. The length is 16 feet longer than the width. The surface area of the pool is 420 square feet. What are the dimensions of the pool? 6.5

10/16/2015Math KM65 Let’s see a Diagram! w w + 16 Area = length x width 420 = (w+16)(w) w w – 420 = 0 (w - 14)(w + 30) = 0 w = 14 or w =

10/16/2015Math KM66 Answer the Question! w w + 16 Pat’s pool is 14 feet wide and 30 feet long. = 14 feet = = 30 feet 6.5

10/16/2015Math KM67 Is it “Square”? Lilly and Mike are building a deck and want to make sure it is “square” (the corners are 90 degrees). If the deck is 12’ by 16’, what diagonal measurement is needed to be sure it is “square”? 12’ 16’ d 6.5

10/16/2015Math KM68 Time for the Pythagorean Equation! 12’ 16’ d 6.5

10/16/2015Math KM69 Solve for d d = -20 or d = 20 If the diagonal is 20’ long, the deck will be “square”. 6.5

10/16/2015Math KM70 That’s All For Now!