Teacher's Page Unit 4.1 in two parts: Part 1: Polynomials and Part 2: Quadratics Benchmarks: A.SSE.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. A.SSE.3a Factor a quadratic expression to reveal the zeros of the function it defines. A.APR.1 1 Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication: add and subtract linear and quadratic polynomials, and multiply binomials. A.CED.1 1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear, quadratic, and exponential functions. Limit exponential numbers to have integer inputs only. A.REI.4 Solve quadratic equations in one variable. A.REI.4b Solve quadratic equations by inspection (eg., For x 2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b. Resources: Chapter 9 in Algebra McDougal Littel Timeline: April 1 (4/1) Day 1: Intro to Polynomials include closure property Day 2: Adding and subtracting polynomials Day 3: Multiplying polynomials Day 4: Quiz 1 on Polynomials...intro to factoring Day 5: Factoring basic forms (4/8) Day 6: Factoring with coefficients Day 7: Special Factoring Forms Day 8: Review/ Quiz 2 on factoring Day 9: Review Unit 4a Day 10: Test Unit 4 1a (4/15) Day 11: Solving Quadratics using Factoring Day 12: Solving Quadratics with Square roots Day 13: Quiz 3 on Solving Intro to completing the Square Day 14: Completing the Square Day 15: Quadratic Formula (no negative roots) (4/22) Day 16: Quadratic Formula (Negative roots) Intro to complex numbers. Day 17: Quiz 3 on Using completing the square and Quadratic Equations Day 18: Review Test on Solving Quadratics Day 19: Test 4 1b Day 20: Intro to Radicals Day 21: Mar 20 8:26 AM 3/25/2014 Guest speaker: Wes Devries from Art Institute Topic: Math as related to the different real life application in careers related to art of different types. Mar 25 1:14 PM 1
WARM UP Combine Like Terms 1. 2x 2 + 3y 4x + 2x + 3x 2 7y March 26, 2014 2. 9 5x + 8y + 2 x Polynomials: What is the definition of the word? An expression that is separated by a mathematical operation. Each expression is called a term. Mar 26 6:17 AM 2
Polynomials Degree Variable 2x 3 + 5x 2 4x + 7 Leading Coefficient TOTAL TERMS: Coefficients Constant Term Do you know all the vocabulary that are associated with polynomials? Polynomial Standard form: Polynomials are written in the order of largest exponent expression term to the constant as the last term. Mar 26 6:27 AM 3
Rewriting Polynomials in Standard Form (Write the terms in decending order of the exponents.) Ex 1. 7 + 4x 3 x 2 Degree: Leading Coefficient: Number of terms: Ex 2. 5x + 7 x 2 + 3x 3 Degree: Leading Coefficient: Number of terms: 4
Polynomial Examples Degree Classified # of Classified by by Degree Terms # of terms 6 0 Constant 1 Monomial 2x 1 Linear 1 Monomial 3x + 1 1 Linear 2 Binomial x 2 +2x 5 2 Quadratic 3 Trinomial 4x 3 8x + 5 3 Cubic 2 Trinomial 2x 4 7x 3 5x +1 4 Quartic 4+ Polynomial Rewriting & Classifying Polynomials in Standard Form Ex 1. 5 + 7x 3 Degree: Leading Coefficient: Number of terms: Classification: 5
Ex 2. 8x + 2 4x 2 + 9x 3 Degree: Leading Coefficient: Number of terms: Classification: 1. 18x + x 2 3x 3 + 7 2. 13 + 4x 3 x 3. 3x 2 + 4x 4. 15 + 7x 3 + x 2 3x 4 5. 7 + x 2 6. 2x + 5x 2 7 7. 9x 3 4 + x 12x 2 8. 90 9. 7x + 14x 3 10. 25x 4 6
March 27, 2014 WARM UP Rewrite in Standard Form 1. 2x + 3 4x 4 + 6x 2 7x 3 2. 9 5x 3 + 8x 4 x Adding Polynomials Ex. 1 (Vertical Form) (5x 3 x + 2x 2 + 7) + ( 3x 2 + 7 4x 3 ) Step 1: Rewrite in Standard Form and Align Terms. 7
Step 2: Add Like Terms. Adding Polynomials Ex. 2 (Horizontal Form) ( 2x 2 + x 5) + ( x + x 2 + 6) Step 1: Organize so Like Terms are Together. 8
Step 2: Add Like Terms. Example: ADDING POLYNOMIALS (6x 3 2x 2 + x + 7) + (4x 2 8 x 3 ) + (2x 2 x 4) 9
Example: ADDING POLYNOMIALS (x 2 4x + 3) + (3x 2 5 3x) Subtracting Polynomials Ex. 1 (Vertical Form) ( 2x 3 x + 5x 2 + 8) ( 3x 4 2x 3 ) Step 1: Keep Change Opposite and Rewrite in Standard Form. 10
Step 2: Align Terms and Add. ( 2x 3 + 5x 2 x + 8) + ( 2x 3 3x + 4 ) Subtracting Polynomials Ex. 2 (Horizontal Form) (x 2 8) ( 7x + 4x 2 ) Step 1: Keep Change Opposite! 11
(x 2 8) + ( 7x 4x 2 ) Step 2: Organize so Like Terms are Together and Add. Example: SUBTRACTING POLYNOMIALS (x 2 4x + 3) (3x 2 5 3x) 12
Example: SUBTRACTING POLYNOMIALS (3x 2 5x + 3) (2x 2 4 x) WARM UP Rewrite in Standard Form 1. 16x + 11 7x 4 + 21x 2 14x 3 March 28, 2014 2. 12 7x 3 + 9x 4 3x 13
ADDING POLYNOMIALS (x 2 4x + 3) + (3x 2 3x 5) SUBTRACTING POLYNOMIALS ( x 2 + 3x 4) (2x 2 + x 1) 14
Rewrite in Standard Form. Classify by degree and # of terms. 2x + 5x 2 1 SUBTRACTING POLYNOMIALS (8 + x 3x 2 ) ( 8x + x 2 + 4) 15
Rewrite in Standard Form. Classify by degree and # of terms. 10 + 12x 3 ADDING POLYNOMIALS (5x 2 1 2x) + ( 3x 2 6x + 4) 16
ADDING POLYNOMIALS (4x 2 5x 9) + ( x 7 5x 2 ) Rewrite in Standard Form. Classify by degree and # of terms. 17x 17
SUBTRACTING POLYNOMIALS (12x 3 + 10) (18x 3 3x 2 + 6) Rewrite in Standard Form. Classify by degree and # of terms. 6x 3x 2 7 + 18x 3 18
ADDING POLYNOMIALS ( 9x 3 + 12 ) + ( 7x + 3 4x 3 ) SUBTRACTING POLYNOMIALS (3x 3 1 7x) ( 7x 2 + 3 2x 3 ) 19
Rewrite in Standard Form. Classify by degree and # of terms. 5x + 7 3x 3 SUBTRACTING POLYNOMIALS (12x 3 + 15 16x) (12x 2 15) 20
WARM UP Combine Like Terms 1. 2x 7 + x 3x 2 + 12 2. 9 5x + 8x 2 + 2x 11x 2 Adding Polynomials (5x 2 + 2x 7) + ( x 3x 2 + 9) 21
Subtracting Polynomials (2x + 3x 2 1) ( 3x 2 + 4 + 5x) Find the Perimeter 3x + 1 3x + 1 2x 2 7 22
Multiplying Polynomials 4x 2 (x 2 3x + 1) Multiplying Polynomials (3x 5)(2x 2 + 8x + 3) 23
Multiplying Polynomials (3x 5)(2x 2 + 8x + 3) Multiplying Polynomials (3x + 2)(2x + 1) 24
April 1, 2014 WARM UP Add or Subtract Polynomials: 1. (2x 3x 2 7) + (x 2 + 6) 2. (9 5x + 8x 2 ) (5x 3x 2 + 7) 25
Multiplying Polynomials (3x 2 + 8x 2)(x 2 2x + 1) Multiplying Polynomials (3x + 8)(x 1) 26
Special Pattern Sum and Difference Pattern (a +b)(a b) = a 2 b 2 (x + 9)(x 9) Special Pattern Sum and Difference Pattern (3x + 7)(3x 7) 27
Special Pattern Square of a Binomial Pattern (a + b) 2 = a 2 + 2ab + b 2 (x + 3) 2 Special Pattern Square of a Binomial Pattern (x 8) 2 28
Multiplying Polynomials (4x 6) 2 3x + 5 Find the Area x + 2 29
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WARM UP Add or Subtract Polynomials: 1. (15x 27x 2 3) + (18x 2 + 12) April 7, 2014 2. (33 11x + 28x 2 ) (15x 31x 2 + 9) PROCEDURES DURING GROUP WORK 31
You are responsible for your own job and the results of the group. In the working world, you are responsible for your own job and the results of the people you work with. If you have a question, ask your support buddies. In the working world, you seek, ask and research because you are expected to act on your own initiative. 32
You must be willing to help if a support buddy asks for help. In the working world, you are expected to apply teamwork skills. If no one can answer a question, agree on a single question and everyone raise their hands for help from the teacher. In the working world, negotiating and reaching agreements are the keys to success. 33
ANY QUESTIONS? IN GROUPS OF 3, YOU WILL WORK ON THE MULTIPLYING POLYNOMIALS SIDE. 34
April 8, 2014 WARM UP Multiplying Polynomials: 1. (12x + 1)(2x 3) 2. ( 4x 6)(x + 5) Multiplying Special Case Polynomials 1. (x + 5)(x 5) 2. (n 1)(n + 1) 35
Multiplying Special Case Polynomials 3. (p 1) 2 4. (x 3)(x + 3) Multiplying Special Case Polynomials 5. (x 4) 2 6. (n + 3) 2 36
Multiplying Special Case Polynomials 7. (x 5)(x + 5) 8. (n 5) 2 Multiplying Special Case Polynomials 9. (2k 2 + 1) 2 10. (8a 2 + 4)(8a 2 4) 37
Multiplying Special Case Polynomials 11. (2 + 5n 2 ) 2 12. (3x 7)(3x + 7) April 9, 2014 WARM UP Multiplying Polynomials: 1. 7x(12x 1) 2. 3( 8x 2 + 11) 38
Homework Questions?? Any Questions before the Quiz??? Unit 4A QUIZ 1 Retake 1. (13x 2x 2 9) + (11x 2 + 1) 2. (37 14x + 22x 2 ) (5x 3x 2 + 19) 3. (12r 3 + 3r 2 + 7) + ( 21r 3 + 4r 6r 2 ) 4. 3x( 6x 5) 5. (2m 1)(7m + 9) 39
Unit 4A QUIZ 1 Retake 6. (3x 6)(5x + 6) 7. (4n 8) 2 8. (3x 2 + 10)(3x 2 10) 9. (5x 2)(5x + 2) 10. (9x + 7) 2 WARM UP 1. (x + 3)(x + 4) 2. (x + 7)(x + 2) April 9, 2014 40
Factoring a Quadratic Trinomial To Factor x 2 + bx + c into two binomials (x + p)(x + q), we must find the values of p and q such that b = p + q and c = pq. Factoring a Quadratic Trinomial Example 1: x 2 + 3x + 2 41
Factoring a Quadratic Trinomial Example 2: x 2 + 8x + 15 Factoring a Quadratic Trinomial Example 3: x 2 + 5x + 6 42
Factoring a Quadratic Trinomial Example 4: x 2 5x + 6 Factoring a Quadratic Trinomial Example 5: x 2 2x 8 43
Factoring a Quadratic Trinomial Example 6: x 2 + 7x 18 Factoring a Quadratic Trinomial Example 7: x 2 + 3x 4 44
Factoring a Quadratic Trinomial Example 8: x 2 7x 30 Factor: x 2 10x 24 A) (x 6)(x + 4) B) (x 4)(x + 6) C) D) (x 2)(x + 12) (x 12)(x + 2) 45
April 10, 2014 WARM UP 1. (9x 3 + 5x 2 + 11) + ( 2x 3 + 9x 8x 2 ) 2. Factor: x 2 + 7x + 12 Factoring a Quadratic Trinomial To Factor ax 2 + bx + c into two binomials (mx + p)(nx + q), we must find the values of m, n, p, and q so that b = mq + pn. 46
Factoring a Quadratic Trinomial Example 1: 2x 2 + 9x + 9 Factoring a Quadratic Trinomial Example 2: 3x 2 + 10x + 3 47
Factoring a Quadratic Trinomial Example 1: 2x 2 + 9x + 9 Factoring a Quadratic Trinomial Example 2: 3x 2 + 10x + 3 48
Factoring a Quadratic Trinomial Example 3: 2x 2 + 7x + 3 Factoring a Quadratic Trinomial Example 4: 3x 2 4x 7 49
Factoring a Quadratic Trinomial Example 5: 2x 2 + 19x 10 Factoring a Quadratic Trinomial Example 6: 6x 2 11x 10 50
Factoring a Quadratic Trinomial Example 7: 3x 2 4x + 7 Which of the following is a factor of: 5x 2 + 23x + 12 A. (5x + 2) B. (5x + 3) C. (5x + 4) D. (5x + 6) 51
WARM UP 1. (8x 2 + 2x 4 + 15x) ( 5x 3 + 6x 8x 2 ) April 10, 2014 2. Factor: x 2 6x 16 If there is a common factor, you must factor it out first before starting the bottoms up method. 6x 2 2x 8 52
Factor out the common factor first! 6x 2 + 9x 27 Factoring a Quadratic Trinomial 1. x 2 + 9x 10 53
Factoring a Quadratic Trinomial 2. 2x 2 9x + 10 Factoring a Quadratic Trinomial 3. x 2 14x 15 54
Factoring a Quadratic Trinomial 4. 4x 2 + 9x 9 Factoring a Quadratic Trinomial 5. x 2 9x + 14 55
Factoring a Quadratic Trinomial 6. x 2 + 13x + 42 Factoring a Quadratic Trinomial 8. x 2 + 26x + 25 56
Factoring a Quadratic Trinomial 10. 10x 2 + 25x 60 April 21, 2014 WARM UP 1. (5x + 7)(5x 7) 2. Factor: x 2 4x + 4 57
Factoring Special Products Difference of Two Squares Pattern: a 2 b 2 = (a + b)(a b) Example: x 2 4 Factoring Special Products Difference of Two Squares Pattern: Example: 9x 2 16 58
Factoring Special Products Difference of Two Squares Pattern: Example: 4x 2 25 Factoring Special Products Perfect Square Trinomial Pattern: a 2 + 2ab + b 2 = (a + b) 2 a 2 2ab + b 2 = (a b) 2 Example: x 2 4x + 4 59
Factoring Special Products Perfect Square Trinomial Pattern: Example: 16x 2 + 24x + 9 Factoring Special Products Perfect Square Trinomial Pattern: Example: 4x 2 + 12x + 9 60
Factoring a Quadratic Trinomial 9) 6n 2 + 5n 6 April 22, 2014 Warm Up 1. (x 3) 2 61
Factoring a Quadratic Trinomial x 2 + 13x + 42 Factoring a Quadratic Trinomial 2x 2 + 21x 11 62
April 23, 2014 QUIZ 2 1. (17x 3x 2 8) + (12x 2 + 13) 2. (2m 1)(7m + 9) Factor: 3. x 2 + 11x + 18 4. x 2 6x + 8 5. 8x 2 + 2x 3 6. 6x 2 + 15x + 6 WARM UP 1. (15x 27x 2 3) + (18x 2 + 12) April 24, 2014 2. (2x + 8)(3x 2 +5x 2) 63
Solving Polynomial Equations in Factored Form (x 2)(x + 3) = 0 Solving Polynomial Equations in Factored Form (2x 3)(x + 5) = 0 64
Solving Polynomial Equations in Factored Form (x 6)(x + 9) = 0 Solving Polynomial Equations in Factored Form (x + 5) 2 = 0 65
Solving Polynomial Equations in Factored Form 4(3x 9)(2x + 4) = 0 Solving Polynomial Equations in Factored Form 3(4x 12)(5x + 10) = 0 66
Solving Polynomial Equations in Factored Form (2x + 1)(3x 2)(x 1) = 0 Which of the following equations has solutions of 7 and 4? A. (x + 7)(x 4) = 0 B. (x 7)(x + 4) = 0 C. (x 7)(x 4) = 0 D. (x + 7)(x + 4) = 0 67
April 25, 2014 WARM UP 1. (12 7x 2 + 14x) (8x 2 12) 2. Factor: (3x 2 +5x 2) Solving Quadratic Equations x 2 + 7x + 10 = 0 68
Solving Quadratic Equations x 2 9x = 14 Solving Quadratic Equations 7x 2 10x + 3 = 0 69
Solving Quadratic Equations 18x 2 12x = 24x 2 What is the solution set for the following equation? x 2 10x + 9 = 0 A. { 9, 1} B. { 9, 1} C. { 1, 9} D. {1, 9} 70
WARM UP 1. (17 5x 2 + 54x) (14x 2 + 8 3x 3 ) April 28, 2014 2. Factor: (8x 2 +2x 3) FACTOR COMPLETELY 18x 4 9x 3 71
Factoring by Grouping Factor x 3 + 2x 2 + 3x + 6 completely. Factoring by Grouping Factor 8x 3 64x 2 + x 8 completely. 72
Factoring by Grouping Factor 12x 3 + 2x 2 30x 5 completely. Factoring by Grouping Factor 12x 3 21x 2 + 28x 49 completely. 73
FACTOR COMPLETELY 9x 2 + 66x + 21 FACTOR COMPLETELY 15x 2 27x 6 74
FACTOR COMPLETELY 7x 3 + 28x 2 21x 75