2. Factoring By Lindsay Hojnowski (2014) Buffalo State College 04/2014L. Hojnowski © 20141 Click here to play tutorial introduction Greatest Common Factor (GCF) Monomial x Polynomial Binomial x Trinomial Trinomial Difference of Perfect Squares (DOPS) Binomial x Binomial

3. Learning Objectives Students will be able to multiply monomials by polynomials 90% of the time. Students will be able to utilize the traditional and the box method to find the products of polynomials with 85% degree of accuracy. The learner will be able to multiply special products. Students will be able to factor using the greatest common factor (GCF) and grouping 85% of the time. 04/2014L. Hojnowski © 20142 Aim for the Target

4. Learning Objectives Continued… Students will be able to factor trinomials (a = 1 and a > 1) with 85% degree of accuracy. Students will be able to factor using the difference of two perfect squares (DOPS) 90% of the time. Students will be able to factor completely 80% of the time. 04/2014L. Hojnowski © 20143 Aim for the Target

5. Menu 04/2014L. Hojnowski © 2014 Vocabulary 4 Multiplying a Monomial and a Polynomial- Example 1 References Question #1 Question #2 Question #3 Question #6 Question #4 Question #5 Question #7 Multiplying a Monomial and a Polynomial- Example 2 and 3 Multiplying Binomials – Two Different Methods Multiplying Binomials – Traditional Distributing Example 1 Multiplying Binomials – Box Method Example 1 Multiplying Binomials – Traditional Distributing Example 2 and 3 Multiplying Binomials – Box Method Example 2 and 3 Multiplying Binomials – Using Two Different Methods Special Products- Example 2 and 3 GCF Factoring- Steps GCF Factoring- Example 2 and 3 Grouping Conditions Grouping- Steps Grouping- Examples Special Products- Example 1 Trinomials (a = 1)- Steps Trinomials (a = 1)- Example 2 and 3 Trinomials (a > 1)- Steps Trinomials (a > 1)- Example 2 and 3 DOPS- Conjugates Conjugate Examples Review of Perfect Squares DOPS- Examples 3 - 6 Factor Completely- Steps Factor Completely- Example 2 Factor Completely- Example 3 and 4 Question #8 Question #9 Question #10 DOPS- Example 1 and 2

6. Vocabulary A.Constant: a number that does not change; it remains the same  Example: any number; 2, -8 B.Variable: a letter that represents a number, it’s value will vary  Example: any letter; x, b, n, etc. C.Term: a variable, constant, or a product of variables and constants  Example: 4x 2, 2xy, y, 4 D.Polynomial: the sum of many terms  Example: 8x 2 + x + 2, -2x 2 + 4x - 3 E.Monomial: a polynomial with 1 term  Example: 4x 2, 6xy, 16 F.Binomial: a polynomial with 2 terms  Example: x 2 + 4, x + 6 G.Trinomial: a polynomial with 3 terms  Example: 7x 2 - x - 18, y 2 + 2y + 1 04/2014L. Hojnowski © 20145 Vocabulary

7. Multiplying a Monomial and a Polynomial- Example 1 04/2014L. Hojnowski © 20146 Reminder: When multiplying variables, add the exponents. **When there is no exponent written, the exponent is 1** Example 1: x (4x 2 + 3x + 2) x (4x 2 + 3x + 2) 4x 3 + 3x 2 + 2x Monomial x Trinomial Distribute Combine like terms (when necessary)

8. Multiplying a Monomial and a Polynomial- Example 2 and 3 04/2014L. Hojnowski © 20147 Example 2: 2/3np 2 (20p 2 + 9n 2 p – 12) (Distribute) 40/3np 4 + 6n 3 p 3 – 8np 2 (Can’t combine) Example 3: 2b (b 2 + 4b + 8) – 3b(3b 2 + 9b + 18) (Distribute) 2b 3 + 8b 2 + 16b – 9b 3 – 27b 2 – 54b (Combine like terms) -7b 3 – 19b 2 + 70b

9. Multiplying Binomials- Two Different Methods Traditional Distributing: (x+1) (x + 2) x (x + 2) + 1 (x + 2)  Distribute x 2 + 2x + 1x + 2  Combine like terms x 2 + 3x + 2 04/2014L. Hojnowski © 20148 Box Method (Modeling): x 2 + 1x + 2x + 2 x 2 + 3x +2 Example: (x+1) (x + 2)

10. Multiplying Binomials- Traditional Distributing Example 1 04/2014L. Hojnowski © 20149 Example 1: (x+ 2) (x + 4) (x+ 2) (x + 4) x (x + 4) + 2 (x + 4) Distribute x 2 + 4x + 2x + 8Combine like terms x 2 + 6x + 8 -To find the product of two binomials- you must distribute each term in the first set of ( ) to each term in the second set of ( )

11. Multiplying Binomials- Traditional Distributing Example 2 and 3 04/2014L. Hojnowski © 201410 Example 2: (2y - 5) (y - 6) (2y - 5) (y - 6) 2y (y - 6) – 5 (y - 6) 2y 2 – 12y – 5y + 30 2y 2 – 17y + 30 -To find the product of two binomials- you must distribute each term in the first set of ( ) to each term in the second set of ( ) Example 3: (3x - y) (4x + 2y) (3x - y) (4x + 2y) 3x (4x + 2y) - y (4x + 2y) 12x 2 + 6xy - 4xy – 2y 2 12x 2 + 2xy – 2y 2 Distribute Combine like terms

12. Multiplying Binomials- Box Method Example 1 04/2014L. Hojnowski © 201411 - Box Method is also known as modeling Example 1: Find the product: (8m – 1)(8m + 1) 64m 2 + 8m – 8m – 1 64m 2 – 1 ** Multiply the product of each box! ** Rewrite the terms in each box and combine like terms

13. Multiplying Binomials- Box Method Example 2 and 3 04/2014L. Hojnowski © 201412 Example 2: Find the product: (k + 4)(5k - 1) 5k 2 + 20k – k – 4 5k 2 + 19k - 4 Example 3: Find the product: (2a - 4)(3a - 6) 6a 2 – 12a– 12a + 24 6a 2 – 24a + 24

14. Multiplying Binomials- Using Two Different Methods 04/2014L. Hojnowski © 201413 Traditional Distributing: (2b + 4) (2b 2 - 8b + 3) 2b (2b 2 - 8b + 3) + 4 (2b 2 - 8b + 3) (Distribute) 4b 3 – 16b 2 + 6b + 8b 2 - 32b + 12 (Combine like terms) 4b 3 – 8b 2 - 26b + 12 Box Method (Modeling): 4b 3 – 16b 2 + 6b + 8b 2 - 32b + 12 4b 3 – 8b 2 - 26b + 12 Find the product using two different methods: (2b + 4) (2b 2 - 8b + 3) Should be the same answer (if they are not you made a mistake)

15. Special Products- Example 1 04/2014L. Hojnowski © 201414 Example 1: (x – 6) 2 (x - 6)(x – 6) x(x – 6) – 6(x – 6) x 2 – 6x - 6x + 36 x 2 – 12x + 36 -Square of sum and differences means write the binomial two times and distribute (using either traditional method or box method) Find each product: Write it twice Multiply/distribute Combine like terms

16. Special Products- Example 2 and 3 04/2014L. Hojnowski © 201415 Example 2: (m 2 – 2) 2 (m 2 – 2)(m 2 – 2) m 2 (m 2 – 2) – 2(m 2 – 2) m 4 – 2m 2 – 2m 2 + 4 m 4 – 4m 2 + 4 Example 3: ((3/4)k + 8) 2 ((3/4)k + 8)((3/4)k + 8) (3/4)k ((3/4)k + 8)+ 8((3/4)k + 8) (9/16)k 2 + 6k + 6k + 64 (9/16)k 2 + 12k + 64 Special Products

17. GCF Factoring- Steps 04/2014L. Hojnowski © 201416 Steps to GCF Factoring STEPS: 1)Find the GCF by taking the lowest exponent (and finding a common factor of the two terms) 2)Write the GCF before parentheses 3)Divide each term of the polynomial by the GCF A)Divide coefficients B)Subtract the exponents Example 1: Factor the following polynomial using GCF 7y 2 – 21y7y 2 – 21y GCF = 7y7y 7y7y ( y – 3) GCF out front

18. GCF Factoring- Example 2 and 3 04/2014L. Hojnowski © 201417 Example 3: Factor the following polynomial using GCF 2x 2 + 4x + 62x 2 + 4x + 6 GCF = 2 2 2 22 (x 2 + 2x + 3) GCF out front Example 2: Factor the following polynomial using GCF 27x 2 – 18x 3 GCF = 9x 2 9x 2 9x 2 9x 2 (3 + 2x) GCF out front

19. Grouping Conditions 04/2014L. Hojnowski © 201418 A polynomial can be factored by grouping ONLY if all of the following conditions exist: 1)There are four or more terms 2)Terms have a common factor that can be grouped together 3)There are 2 common factors that are identical to each other (the parentheses match) Grouping Example

20. Grouping- Steps 04/2014L. Hojnowski © 201419 Steps to factoring by grouping STEPS: 1)Group the terms with common factors 2)Factor the GCF from each group 3)Rewrite the final answer as a (binomial)(binomial) Directions: Factor the following polynomial. Show all your work. Example 1: 4qr + 8r + 3q +6 4r (q + 3) + 3 (q + 3) (4r + 3) (q + 3) GCF = 4r GCF = 3 These should match

21. Grouping- Examples 04/2014L. Hojnowski © 201420 Example 2: 3x 3 – 6x 2 + x - 2 GCF = 3x 2 GCF = 1 3x 2 (x - 2) + 1 (x - 2) These should match (3x 2 + 1) (x - 2) Directions: Factor the following polynomials. Show all your work. Example 3: 2mk – 12m – 7k + 42 GCF = 2m GCF = -7 2m (k - 6) - 7(k - 6) These should match (2m - 7) (k - 6)

22. Trinomial (a = 1)- Steps 04/2014L. Hojnowski © 201421 Steps to Factoring a=1 Trinomials STEPS: 1)When the leading coefficient is 1, ask yourself “what numbers multiply to the last term and adds to the middle term?” 2)Rewrite the trinomial as a polynomial with 4 terms (the middle term will get replaced by the 2 new terms that add to it) 3)Factor the polynomial by grouping Example 1: x 2 + 10x + 24x 2 + 10x + 24 Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24x 2 + 4x + 6x + 24 4, 6 are factors of 24 that add to 10 x (x + 4) + 6 (x + 4) (x + 6)(x + 4)

23. Trinomial (a = 1)- Example 2 and 3 04/2014L. Hojnowski © 201422 Example 2: x 2 – 2x - 63 Factors of -63: 1, 3, 7, 9, 21, 63 (one has to be negative) 7, -9 are factors of -63 that add to -2 x 2 – 2x - 63 x 2 + 7x - 9x – 63 x (x + 7) - 9 (x + 7) (x - 9)(x + 7) Directions: Factor the following polynomials. Show all your work. Example 3: x 2 + x - 56 Factors of -56: 1, 2, 4, 7, 8, 14, 28, 56 (one has to be negative) 8, -7 are factors of -56 that add to 1 x 2 + x - 56 x 2 + 8x - 7x – 56 x (x + 8) - 7 (x + 8) (x - 7)(x + 8)

24. Trinomial (a > 1)- Steps 04/2014L. Hojnowski © 201423 STEPS: 1)First see if a GCF can be factored out (this is ALWAYS the 1 st step of factoring) 2)Find the product of a and c from the trinomial (make sure you include the sign of each number) 3)Think of a pair of numbers whose sum is equal to ac 4)Break up the middle term into those two numbers 5)GCF Factor twice (grouping) Recall: Standard from of a quadratic equation is y = ax 2 + bx + c Example 1: 2x 2 + 5x + 3 a = 2 c = 3, ac = 6 Factors of 6: 1, 2, 3, 6 2, 3 are factors of 6 that add to 5 2x 2 + 2x + 3x + 3 2x (x + 1) + 3 (x + 1) (2x + 3)(x + 1)

25. Trinomial (a > 1)- Example 2 and 3 04/2014L. Hojnowski © 201424 Example 2: 4x 2 - 12x + 5 a = 4 c = 5, ac = 20 Factors of 20: 1, 2, 4, 5, 10, 20 -2, -10 are factors of 20 that add to -12 4x 2 - 2x – 10x + 5 2x (2x - 1) - 5 (2x - 1) (2x - 5)(2x - 1) Example 3: 3x 2 + 17x + 10 a = 3 c = 10, ac = 30 Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30 2, 15 are factors of 30 that add to 17 3x 2 + 2x + 15x + 10 x (3x + 2) + 5 (3x + 2) (x + 5)(3x + 2)

26. DOPS- Conjugates 04/2014L. Hojnowski © 201425 When two binomials look the same, but have different signs between the two terms, the binomials are called conjugates Conjugates

27. Conjugate Examples 04/2014L. Hojnowski © 201426 For each of the following binomials, write the conjugate: Expression: x 2 – 3 Conjugate:x 2 + 3 Expression: a + b Conjugate:a + b The conjugate can be very useful because when you multiply something by its conjugate, you get squares like this… (a + b)(a – b) = a 2 – b 2

28. DOPS- Example 1 and 2 04/2014L. Hojnowski © 201427 Directions: Multiply each of the following. Example 1: (y – 5) (y + 5) y 2 - 25 Example 2: (8x - y) (8x + y) 64x 2 – y 2 This expression is called DOPS (Difference of Perfect Squares)- **Notice the minus sign in between the terms and that both terms are perfect squares)

29. Review of Perfect Squares 04/2014L. Hojnowski © 201428 Let’s review the perfect squares 1- 15: Perfect Square

30. DOPS- Examples 3 - 6 04/2014L. Hojnowski © 201429 Directions: Factor each of the following. Example 3: x 2 - 81 (x – 9) (x + 9) Example 5: 16h 2 – 9a 2 (4h – 3a)(4h + 3a) Example 4: 64 - y 2 (8 – y) (8+ y) Example 6: 100g 2 –h 2 (10g – h)(10g + h) DOPS

31. Factoring Completely- Steps 04/2014L. Hojnowski © 201430 Steps to Factoring Completely STEPS: 1)Factor out the GCF (if there is one) 2)Factor the polynomial (DOPS, Trinomial (a = 1 or ac method), Grouping) Example 1: Factor completely 2x 2 + 4x - 16 GCF = 2 2 2 22 (x 2 + 2x - 8 ) GCF out front 2 (x 2 + 2x - 8)  you can factor what is in the parentheses (Trinomial) Factors of -8: 1, 2, 4, 8 (one has to be negative)  4, -2 are factors of -8 that add to 2 x 2 + 4x – 2x - 8 x (x + 4) - 2 (x + 4) 2(x - 2)(x + 4) Don’t forget the GCF out front!

32. Factoring Completely- Example 2 04/2014L. Hojnowski © 201431 Example 2: Factor completely 10y 3 – 35y 2 + 30y GCF = 5y 5y 5y 5y5y (2y 2 – 7y + 6) GCF out front 5y (2y 2 – 7y + 6)  you can factor what is in the parentheses (Trinomial ac) a = 2 c = 6, ac = 12  Factors of 12: 1, 2, 3, 4, 6, 12  -3, -4 are factors of 12 that add to -7 2y 2 – 4y – 3y + 6 5y(2y - 3)(y - 2) 2y (y - 2) - 3 (y - 2) Don’t forget the GCF out front!

33. Factoring Completely- Example 3 and 4 Example 3: Factor completely 27g 3 – 3g GCF = 3g 27g 3 – 3g 3g 3g 3g (9g 2 – 1) GCF out front 3g (9g 2 – 1)  Factor parentheses (DOPS) 3g (3g – 1) (3g + 1) 04/2014L. Hojnowski © 201432 Example 4: Factor completely 2y 4 – 50 GCF = 2 2y 4 – 50 2 2 2 (y 4 – 25) GCF out front 2 (y 4 – 25)  Factor parentheses (DOPS) 2 (y 2 – 5) (y 2 + 5)

34. Quiz Question #1 04/2014L. Hojnowski © 201433 1.Simplify: (3/4)m 2 n (16m 3 n 2 – 4m 2 n 3 + 6mn) a. a. 16m 6 n 2 – 3m 4 n 3 + 6m 2 n b. 12m 5 n 3 + 3m 3 n 3 + (9/2)m 3 n 2 b. c. c. 12m 6 n 2 + 3m 4 n 3 + 6m 2 n d. 12m 5 n 3 – 3m 4 n 4 + (9/2)m 3 n 2d.

35. Try Again… 04/2014L. Hojnowski © 201434 When distributing, you forgot to multiply the fraction and 16. Also, when you multiply variables, add the exponents! Quiz Question #1 Quiz Question #2 Try Again

36. Try Again… 04/2014L. Hojnowski © 201435 Quiz Question #1 Quiz Question #2 Try Again Be careful of your signs!

37. Try Again… 04/2014L. Hojnowski © 201436 Quiz Question #1 Quiz Question #2 Try Again When you multiply variables, add the exponents! Be careful of your signs and multiplication.

38. Correct!! 04/2014L. Hojnowski © 201437 You distributed correctly. You’re signs are perfect. Great job! Quiz Question #1 Quiz Question #2 Smile

39. Quiz Question # 2 04/2014L. Hojnowski © 201438 2. Find the product of (4x – 3y)(3x + 2y) a.a. 12x 2 – 6y 2 b. 12x 2 – 9xy - 6y 2 c. 12x 2 – 9xy + 6y 2 d. 12x 2 – xy - 6y 2b. c. d.

40. Try Again… 04/2014L. Hojnowski © 201439 These binomials are not conjugates of each other, thus you have to multiply each and every term. Quiz Question #1 Quiz Question #2 Try Again Quiz Question #3

41. Try Again… 04/2014L. Hojnowski © 201440 Try Again Quiz Question #1 Quiz Question #2 Quiz Question #3 Careful when you combine like terms. Go back and check your work.

42. Try Again… 04/2014L. Hojnowski © 201441 Try Again Quiz Question #1 Quiz Question #2 Quiz Question #3 Be careful of your signs and check your answer when you combine like terms.

43. Correct!! 04/2014L. Hojnowski © 201442 Smile Quiz Question #1 Quiz Question #2 Quiz Question #3 You multiplied correctly! Awesome job keeping track of your signs and combining like terms.

44. Quiz Question # 3 04/2014L. Hojnowski © 201443 3. What is the product of ((2/5)y – 4) 2 ? a.a. (4/10)y 2 – (8/5)y - 16b. (4/10)y 2 – (16/5)y + 16b. c. c. (4/25)y 2 – (16/5)y + 16 d. (4/25)y 2 - 16 d.

45. Try Again… 04/2014L. Hojnowski © 201444 Quiz Question #2 Quiz Question #1 Try Again Quiz Question #4 Quiz Question #3 Double check your multiplication. A negative number times a negative number is a POSITIVE number. Also, be sure you multiplied every term.

46. Try Again… 04/2014L. Hojnowski © 201445 Try Again Quiz Question #2 Quiz Question #1 Quiz Question #4 Quiz Question #3 Double check your multiplication. Be sure you are not adding.

47. Correct!! 04/2014L. Hojnowski © 201446 Smile Quiz Question #2 Quiz Question #1 Quiz Question #4 Quiz Question #3 You multiplied correctly! Awesome job keeping track of your signs and combining like terms.

48. Try Again… 04/2014L. Hojnowski © 201447 Try Again Be careful, you can’t just square both terms. You have to write what is in the parentheses twice, and then distribute. Also, a negative number times a negative number is a POSITIVE number. Quiz Question #2 Quiz Question #1 Quiz Question #4 Quiz Question #3

49. Quiz Question # 4 04/2014L. Hojnowski © 201448 4. What is the correct factoring of 4k 3 + 6k 2 + 2k? a. a. 2k (2k 2 + 3k) b. 2k (2k 2 + 3k + 1) c. 2 (2k 3 + 6k 2 + k) d. 4k (k 2 + 2k + 1)b. c. d.

50. Try Again… 04/2014L. Hojnowski © 201449 Careful when you take out the GCF. Try to multiply it out again to see what you’re answer will be. This might help you find your mistake. Try Again Quiz Question #2 Quiz Question #1 Quiz Question #4 Quiz Question #3 Quiz Question #5

51. Correct!! 04/2014L. Hojnowski © 201450 Smile Quiz Question #2 Quiz Question #1 Quiz Question #4 Quiz Question #3 Quiz Question #5 You found the correct GCF and divided the GCF out perfectly. Great job remembering the 1.

52. Try Again… 04/2014L. Hojnowski © 201451 Try Again Quiz Question #2 Quiz Question #1 Quiz Question #4 Quiz Question #3 Quiz Question #5 Careful with your GCF. What else do the terms have in common?

53. Try Again… 04/2014L. Hojnowski © 201452 Try Again Quiz Question #2 Quiz Question #1 Quiz Question #4 Quiz Question #3 Quiz Question #5 Careful with your GCF. Is 4k a factor of each and every one of the terms? Also, make sure you are dividing the GCF out not subtracting.

54. Quiz Question # 5 04/2014L. Hojnowski © 201453 5. Factor: 3dt – 21d – 5t + 35 a.a. (3d + 5)(t – 7)b. (3d - 5)(t – 7)c. (3d + 5)(t + 7)d. (3d - 5)(t + 7)b. c. d.

55. Try Again… 04/2014L. Hojnowski © 201454 Careful when you take out the GCF of the second group. To check your work, you could multiply it out to see what your mistake could be. Try Again Quiz Question #2 Quiz Question #1 Quiz Question #4 Quiz Question #3 Quiz Question #5 Quiz Question #6

56. Correct!! 04/2014L. Hojnowski © 201455 Smile Quiz Question #2 Quiz Question #1 Quiz Question #4 Quiz Question #3 Quiz Question #5 Quiz Question #6 Awesome job factoring by grouping!

57. Try Again… 04/2014L. Hojnowski © 201456 Try Again Quiz Question #2 Quiz Question #1 Quiz Question #4 Quiz Question #3 Quiz Question #5 Quiz Question #6 Careful when you divide out the GCF from both groups. More specifically, be careful of your signs!

58. Try Again… 04/2014L. Hojnowski © 201457 Try Again Quiz Question #2 Quiz Question #1 Quiz Question #4 Quiz Question #3 Quiz Question #5 Quiz Question #6 Careful when you divide out the GCF. More specifically, be careful of your signs!

59. Quiz Question # 6 6. Factor: a 2 + 8a - 48 a. a. (a + 3)(a – 16) b. (a - 3)(a + 16) c. (a + 4)(a + 12) d. (a - 4)(a + 12) b. c. d. 04/2014L. Hojnowski © 201458

60. Try Again… 04/2014L. Hojnowski © 201459 Try Again Quiz Question #2 Quiz Question #1 Quiz Question #4 Quiz Question #3 Quiz Question #5 Quiz Question #6 Quiz Question #7 3 times -16 is -48, but does -16 + 3 = +8? Try to find factors that multiply to -48 that add to 8.

61. Try Again… 04/2014L. Hojnowski © 201460 Try Again Quiz Question #2 Quiz Question #1 Quiz Question #4 Quiz Question #3 Quiz Question #5 Quiz Question #6 Quiz Question #7 -3 times 16 is -48, but does -3 + 16 = +8? Try to find factors that multiply to -48 that add to 8.

62. Try Again… 04/2014L. Hojnowski © 201461 Try Again Quiz Question #2 Quiz Question #1 Quiz Question #4 Quiz Question #3 Quiz Question #5 Quiz Question #6 Quiz Question #7 4 times 12 is 48, but we want -48. Try to find factors that multiply to -48 that add to 8. You’re factors add to 16 right now.

63. Correct!! 04/2014L. Hojnowski © 201462 Smile Quiz Question #2 Quiz Question #1 Quiz Question #4 Quiz Question #3 Quiz Question #5 Quiz Question #6 Quiz Question #7 Awesome job finding factors and having the correct signs!

64. Quiz Question # 7 7. Factor: 3x 2 – 17x + 20 a. a. (3x + 4)(x – 5) b. (3x - 5)(x – 4) c. (3x + 5)(x + 4) d. (3x - 4)(x - 5)b. c. d. 04/2014L. Hojnowski © 201463

65. Try Again… 04/2014L. Hojnowski © 201464 Multiply out your answer. Compare it with the original question. This might help you find your mistake. Try Again Quiz Question #2 Quiz Question #1 Quiz Question #4 Quiz Question #3 Quiz Question #5 Quiz Question #6 Quiz Question #7

66. Correct!! 04/2014L. Hojnowski © 201465 Smile Quiz Question #2 Quiz Question #1 Quiz Question #4 Quiz Question #3 Quiz Question #5 Quiz Question #6 Quiz Question #7 You factored correctly! Great!

67. Try Again… 04/2014L. Hojnowski © 201466 Try Again Quiz Question #2 Quiz Question #1 Quiz Question #4 Quiz Question #3 Quiz Question #5 Quiz Question #6 Quiz Question #7 Multiply out your answer. Compare it with the original question. This might help you find your mistake.

68. Try Again… 04/2014L. Hojnowski © 201467 Try Again Quiz Question #2 Quiz Question #1 Quiz Question #4 Quiz Question #3 Quiz Question #5 Quiz Question #6 Quiz Question #7 Multiply out your answer. Compare it with the original question. This might help you find your mistake.

69. Quiz Question # 8 8. Factor, completely when necessary. 4a 2 - 25 a. a. (4a – 5)(4a + 5) b. (2a + 5)(2a + 5) c. (2a – 5)(2a + 5) d. (4a – 25)(4a + 25)b. c. d. 04/2014L. Hojnowski © 201468

70. Try Again… 04/2014L. Hojnowski © 201469 Be careful, if this is a DOPS question you need to take the square root of both terms in the binomial. Try Again Quiz Question #2 Quiz Question #1 Quiz Question #4 Quiz Question #3 Quiz Question #5 Quiz Question #6 Quiz Question #7 Quiz Question #8

71. Try Again… 04/2014L. Hojnowski © 201470 Try Again Quiz Question #2 Quiz Question #1 Quiz Question #4 Quiz Question #3 Quiz Question #5 Quiz Question #6 Quiz Question #7 Quiz Question #8 Be careful to be a difference of perfect squares you need one – and one +.

72. Correct!! 04/2014L. Hojnowski © 201471 Smile You factored correctly! There is no GCF to be taken out. Quiz Question #2 Quiz Question #1 Quiz Question #4 Quiz Question #3 Quiz Question #5 Quiz Question #6 Quiz Question #7 Quiz Question #8

73. Try Again… 04/2014L. Hojnowski © 201472 Try Again Careful 4 is not a factor of 25, thus the GCF is not 4. Quiz Question #2 Quiz Question #1 Quiz Question #4 Quiz Question #3 Quiz Question #5 Quiz Question #6 Quiz Question #7 Quiz Question #8

74. Quiz Question # 9 9. Factor, completely when necessary. 9m 2 - 144 a. a. (3m – 12)(3m + 12) b. (3m + 12)(3m + 12) c. 9(m 2 - 16) d. 9(m – 4)(m + 4)b. c. d. 04/2014L. Hojnowski © 201473

75. Try Again… 04/2014L. Hojnowski © 201474 Be careful. This does look like a difference of two perfect squares, but your first thought should always be GCF. Try Again Quiz Question #2 Quiz Question #1 Quiz Question #4 Quiz Question #3 Quiz Question #5 Quiz Question #6 Quiz Question #7 Quiz Question #8 Quiz Question #9

76. Try Again… 04/2014L. Hojnowski © 201475 Try Again Quiz Question #2 Quiz Question #1 Quiz Question #4 Quiz Question #3 Quiz Question #5 Quiz Question #6 Quiz Question #7 Quiz Question #8 Quiz Question #9 Be careful. This does look like a difference of two perfect squares, but your first thought should always be GCF. Also, to be a difference of perfect squares you need one – and one +.

77. Try Again… 04/2014L. Hojnowski © 201476 Try Again Quiz Question #2 Quiz Question #1 Quiz Question #4 Quiz Question #3 Quiz Question #5 Quiz Question #6 Quiz Question #7 Quiz Question #8 Quiz Question #9 Be careful. Is your answer factored completely? What does it mean to be factored completely?

78. Correct!! 04/2014L. Hojnowski © 201477 Smile Great job taking out the GCF and factoring what was left correctly! Quiz Question #2 Quiz Question #1 Quiz Question #4 Quiz Question #3 Quiz Question #5 Quiz Question #6 Quiz Question #7 Quiz Question #8 Quiz Question #9

79. Quiz Question # 10 10. Factor, completely when necessary. 25b 3 + 20b 2 – 5b a. a. 5b (5b – 1)(b + 1) b. (5b – 1)(b + 1) c. 5b (5b 2 + 4b - 1) d. 5b (5b + 1)(b - 1)b. c. d. 04/2014L. Hojnowski © 201478

80. Correct!! 04/2014L. Hojnowski © 201479 Smile Great job dividing out the GCF and factoring what was left! Quiz Question #2 Quiz Question #1 Quiz Question #4 Quiz Question #3 Quiz Question #5 Quiz Question #6 Quiz Question #7 Quiz Question #8 Quiz Question #9 Quiz Question #10

81. Try Again… 04/2014L. Hojnowski © 201480 You factored correctly, but you forgot one thing. What do you have to remember to bring down? Try Again Quiz Question #2 Quiz Question #1 Quiz Question #4 Quiz Question #3 Quiz Question #5 Quiz Question #6 Quiz Question #7 Quiz Question #8 Quiz Question #9 Quiz Question #10

82. Try Again… 04/2014L. Hojnowski © 201481 Try Again Your GCF is correct and you divided it out correctly. What does it mean to factor completely? Quiz Question #2 Quiz Question #1 Quiz Question #4 Quiz Question #3 Quiz Question #5 Quiz Question #6 Quiz Question #7 Quiz Question #8 Quiz Question #9 Quiz Question #10

83. Try Again… 04/2014L. Hojnowski © 201482 Try Again Careful of your signs when you factor! Quiz Question #2 Quiz Question #1 Quiz Question #4 Quiz Question #3 Quiz Question #5 Quiz Question #6 Quiz Question #7 Quiz Question #8 Quiz Question #9 Quiz Question #10

84. References McGraw-Hill Companies. (2014). Glencoe Algebra 1 Common Core Edition. New York: McGraw Hill. Seminars.usb.ac.ir. (2011). Hitting the objectives, Retrieved on September 14 th, 2012, from http://www.teambuildinggames.org/role-of-the-team-building- facilitator. http://www.teambuildinggames.org/role-of-the-team-building- facilitator Smiley Face, Retrieved on September 14 th, 2012, from http://ed101.bu.edu/StudentDoc/current/ED101fa10/rajensen/ima ges/happy-face1.png. http://ed101.bu.edu/StudentDoc/current/ED101fa10/rajensen/ima ges/happy-face1.png Wee, E. (2011). Try again, Retrieved on September 15 th, 2012, from http://radionjournals.blogspot.com/2011/04/try-again-part- 3-caring-for-children.html.http://radionjournals.blogspot.com/2011/04/try-again-part- 3-caring-for-children.html 04/2014L. Hojnowski © 201483 Reference from the dictionary