1. Chapter 6 Review Polynomials

2. 2 Practice Product of Powers Property: Try:

3. 3 Answers To Practice Problems 1.Answer: 2.Answer:

4. 4 Practice Using the Power of a Power Property 1.Try: 2.Try:

5. 5 Answers to Practice Problems 1.Answer: 2.Answer:

6. 6 Practice Power of a Product Property 1.Try: 2.Try:

7. 7 Answers to Practice Problems 1.Answer: 2.Answer:

8. 8 Practice Making Negative Exponents Positive 1.Try: 2.Try:

9. 9 Answers to Negative Exponents Practice 1.Answer: 2.Answer:

10. 10 Practice Rewriting the Expressions with Positive Exponents: 1.Try: 2.Try:

11. 11 Answers 1.Answer 2.Answer

12. 12 Practice Quotient of Powers Property 1.Try: 2.Try:

13. 13 Answers 1.Answer: 2.Answer:

14. Let ’ s Try Some Hint: convert to a fraction rather than a decimal! Answers are on the next slide!!!

15. Let ’ s Try Some

16. Monomials - a number, a variable, or a product of a number and one or more variables. 4x, 20x 2 yw 3, -3, a 2 b 3, and 3yz are all monomials. Polynomials – one or more monomials added or subtracted 4x + 6x 2, 20xy - 4, and 3a 2 - 5a + 4 are all polynomials. Vocabulary

17. Like Terms Like Terms refers to monomials that have the same variable(s) but may have different coefficients. The variables in the terms must have the same powers. Which terms are like? 3a 2 b, 4ab 2, 3ab, -5ab 2 4ab 2 and -5ab 2 are like. Even though the others have the same variables, the exponents are not the same. 3a 2 b = 3aab, which is different from 4ab 2 = 4abb.

18. Like Terms Constants are like terms. Which terms are like? 2x, -3, 5b, 0 -3 and 0 are like. Which terms are like? 3x, 2x 2, 4, x 3x and x are like. Which terms are like? 2wx, w, 3x, 4xw 2wx and 4xw are like.

19. A polynomial with only one term is called a monomial. A polynomial with two terms is called a binomial. A polynomial with three terms is called a trinomial. Identify the following polynomials: Classifying Polynomials PolynomialDegree Classified by degree Classified by number of terms 6 –2 x 3x + 1 –x 2 + 2 x – 5 4x 3 – 8x 2 x 4 – 7x 3 – 5x + 1 0 1 1 4 2 3 constant linear quartic quadratic cubic monomial binomial polynomial trinomial binomial

20. Add: (x 2 + 3x + 1) + (4x 2 +5) Step 1: Underline like terms: Step 2: Add the coefficients of like terms, do not change the powers of the variables: Adding Polynomials (x 2 + 3x + 1) + (4x 2 +5) Notice: ‘3x’ doesn’t have a like term. (x 2 + 4x 2 ) + 3x + (1 + 5) 5x 2 + 3x + 6

21. Some people prefer to add polynomials by stacking them. If you choose to do this, be sure to line up the like terms! Adding Polynomials (x2 (x2 + 3x 3x + 1) + (4x 2 +5) 5x 2 + 3x + 6 (x 2 + 3x + 1) + (4x 2 +5) Stack and add these polynomials: (2a 2 +3ab+4b 2 ) + (7a2+ab+-2b 2 ) (2a 2 +3ab+4b 2 ) + (7a2+ab+-2b 2 ) (2a 2 + 3ab + 4b 2 ) + (7a 2 + ab + -2b 2 ) 9a 2 + 4ab + 2b 2

22. Adding Polynomials Add the following polynomials; you may stack them if you prefer:

23. Subtract: (3x 2 + 2x + 7) - (x 2 + x + 4) Subtracting Polynomials Step 1: Change subtraction to addition ( Keep-Change-Change. ). Step 2: Underline OR line up the like terms and add. (3x 2 + 2x + 7) + (- x 2 + - x + - 4) (3x 2 + 2x + 7) + (- x 2 + - x + - 4) 2x 2 + x + 3

24. Subtracting Polynomials Subtract the following polynomials by changing to addition (Keep-Change-Change.), then add:

25. 1. Add the following polynomials: (9y - 7x + 15a) + (-3y + 8x - 8a) Combine your like terms. 6y + x + 7a

26. Combine your like terms. 3a 2 + 7ab + 5b 2 2. Add the following polynomials: (3a 2 + 3ab - b 2 ) + (4ab + 6b 2 )

27. Combine your like terms. x 2 - 3xy + 5y 2 3. Add the following polynomials (4x 2 - 2xy + 3y 2 ) + (-3x 2 - xy + 2y 2 )

28. Rewrite subtraction as adding the opposite. (9y - 7x + 15a) + (+ 3y - 8x + 8a) Combine the like terms. 9y + 3y - 7x - 8x + 15a + 8a 12y - 15x + 23a 4. Subtract the following polynomials: (9y - 7x + 15a) - (-3y + 8x - 8a)

29. Rewrite subtraction as adding the opposite. (7a - 10b) + (- 3a - 4b) Combine the like terms. 7a - 3a - 10b - 4b 4a - 14b 5. Subtract the following polynomials: (7a - 10b) - (3a + 4b)

30. Distribute your negative, and combine like terms 7x 2 - xy + y 2 6. Subtract the following polynomials (4x 2 - 2xy + 3y 2 ) - (-3x 2 - xy + 2y 2 )

31. Find the sum or difference. (5a – 3b) + (2a + 6b) 1.3a – 9b 2.3a + 3b 3.7a + 3b 4.7a – 3b

32. Find the sum or difference. (5a – 3b) – (2a + 6b) 1.3a – 9b 2.3a + 3b 3.7a + 3b 4.7a – 9b

33. Find the sum. Write the answer in standard format. (5x 3 – x + 2 x 2 + 7) + (3x 2 + 7 – 4 x) + (4x 2 – 8 – x 3 ) Adding Polynomials SOLUTION Vertical format: Write each expression in standard form. Align like terms. 5x 3 + 2 x 2 – x + 7 3x 2 – 4 x + 7 – x 3 + 4x 2 – 8 + 4x 3 + 9x 2 – 5x + 6

34. Find the sum. Write the answer in standard format. (2 x 2 + x – 5) + (x + x 2 + 6) Adding Polynomials SOLUTION Horizontal format: Add like terms. (2 x 2 + x – 5) + (x + x 2 + 6) =(2 x 2 + x 2 ) + (x + x) + (–5 + 6) =3x 2 + 2 x + 1

35. Find the difference. (3x 2 – 5x + 3) – (2 x 2 – x – 4) Subtracting Polynomials SOLUTION (3x 2 – 5x + 3) – (2 x 2 – x – 4) = (3x 2 – 5x + 3) + (–1)(2 x 2 – x – 4) = x 2 – 4x + 7 = (3x 2 – 5x + 3) – 2 x 2 + x + 4 = (3x 2 – 2 x 2 ) + (– 5x + x) + (3 + 4)

36. Multiplying Polynomials Distribute

37. Polynomials * Polynomials Multiplying a Polynomial by another Polynomial requires more than one distributing step. Multiply: (2a + 7b)(3a + 5b) Distribute 2a(3a + 5b) and distribute 7b(3a + 5b): 6a 2 + 10ab 21ab + 35b 2 Then add those products, adding like terms: 6a 2 + 10ab + 21ab + 35b 2 = 6a 2 + 31ab + 35b 2

38. Polynomials * Polynomials An alternative is to stack the polynomials and do long multiplication. (2a + 7b)(3a + 5b) 6a 2 + 10ab 21ab + 35b 2 (2a + 7b) x (3a + 5b) Multiply by 5b, then by 3a: (2a + 7b) x (3a + 5b) When multiplying by 3a, line up the first term under 3a. + Add like terms: 6a 2 + 31ab + 35b 2

39. Polynomials * Polynomials Multiply the following polynomials: (x + 5) x (2x + -1) -x + -5 2x 2 + 10x + 2x 2 + 9x + -5 (3w + -2) x (2w + -5) -15w + 10 6w 2 + -4w + 6w 2 + -19w + 10

40. Polynomials * Polynomials (2a 2 + a + -1) x (2a 2 + 1) 2a 2 + a + -1 4a 4 + 2a 3 + -2a 2 + 4a 4 + 2a 3 + a + -1

41. There is an acronym to help us remember how to multiply two binomials without stacking them. Multiply please: (2x + -3)(4x + 5) (2x + -3)(4x + 5) = 8x 2 + 10x + -12x + -15 = 8x 2 + -2x + -15

42. The next slide has all the answers Try multiplying these, 1) (3a + 4)(2a + 1) 2) (x + 4)(x - 5) 3) (x + 5)(x - 5) 4) (c - 3)(2c - 5) (2w + 3)(2w - 3)

43. Answers 1) (3a + 4)(2a + 1) = 6a 2 + 3a + 8a + 4 = 6a 2 + 11a + 4 2) (x + 4)(x - 5) = x2 x2 + -5x + 4x + -20 = x2 x2 + -1x + -20 3) (x + 5)(x - 5) = x2 x2 + -5x + 5x + -25 = x2 x2 + 4) (c - 3)(2c - 5) = 2c 2 + -5c + -6c + 15 = 2c 2 + -11c + 15 5) (2w + 3)(2w - 3) = 4w 2 + -6w + 6w + -9 = 4w 2 + -9

44. 1) Multiply. (2x + 3)(5x + 8) Using the distributive property, multiply 2x(5x + 8) + 3(5x + 8). 10x 2 + 16x + 15x + 24 Combine like terms. 10x 2 + 31x + 24

45. Multiply (y + 4)(y – 3) 1.y 2 + y – 12 2.y 2 – y – 12 3.y 2 + 7y – 12 4.y 2 – 7y – 12 5.y 2 + y + 12 6.y 2 – y + 12 7.y 2 + 7y + 12 8.y 2 – 7y + 12

46. Multiply (2a – 3b)(2a + 4b) 1.4a 2 + 14ab – 12b 2 2.4a 2 – 14ab – 12b 2 3.4a 2 + 8ab – 6ba – 12b 2 4.4a 2 + 2ab – 12b 2 5.4a 2 – 2ab – 12b 2

47. 5) Multiply (2x - 5)(x 2 - 5x + 4) You must use the distributive property. 2x(x 2 - 5x + 4) - 5(x 2 - 5x + 4) 2x 3 - 10x 2 + 8x - 5x 2 + 25x - 20 Group and combine like terms. 2x 3 - 10x 2 - 5x 2 + 8x + 25x - 20 2x 3 - 15x 2 + 33x - 20

48. Multiply (2p + 1)(p 2 – 3p + 4) 1.2p 3 + 2p 3 + p + 4 2.y 2 – y – 12 3.y 2 + 7y – 12 4.y 2 – 7y – 12

49. Example:(x – 6)(2x + 1) x(2x)+ x(1)– (6)2x– 6(1) 2x 2 + x – 12x – 6 2x 2 – 11x – 6

50. 2x 2 (3xy + 7x – 2y) 2x 2 (3xy) + 2x 2 (7x) + 2x 2 (–2y) 2x 2 (3xy + 7x – 2y) 6x 3 y + 14x 2 – 4x 2 y

51. (x + 4)(x – 3) (x + 4)(x – 3) x(x) + x(–3) + 4(x) + 4(–3) x2 x2 – 3x + 4x – 12 x2 x2 + x –

52. (2y – 3x)(y – 2) (2y – 3x)(y – 2) 2y(y) + 2y(–2) + (–3x)(y) + (–3x)(–2) 2y 2 – 4y – 3xy + 6x

53. Multiply (2a + 3) 2 1.4a 2 – 9 2.4a 2 + 9 3.4a 2 + 36a + 9 4.4a 2 + 12a + 9

54. Multiply (x – y) 2 1.x 2 + 2xy + y 2 2.x 2 – 2xy + y 2 3.x 2 + y 2 4.x 2 – y 2

55. 6) Multiply: (y – 2)(y + 2) (y) 2 – (2) 2 y 2 – 4 7) Multiply: (5a + 6b)(5a – 6b) (5a) 2 – (6b) 2 25a 2 – 36b 2

56. Multiply (4m – 3n)(4m + 3n) 1.16m 2 – 9n 2 2.16m 2 + 9n 2 3.16m 2 – 24mn - 9n 2 4.16m 2 + 24mn + 9n 2

57. Simplify. 1) 2)

58. Difference of Squares. Multiply. 1) 2) 3) 4) Inner and Outer terms cancel!

59. Lesson Quiz: Part I 1. A square foot is 3 –2 square yards. Simplify this expression. Simplify. 2. 2 –6 3. (–7) –3 4. 6 0 5. –11 2 1 –121

60. Lesson Quiz: Part II Evaluate each expression for the given value(s) of the variables(s). 6. x –4 for x =10 7. for a = 6 and b = 3

61. Lesson Quiz: Part I Simplify each expression. 1. 2. 3. 4. 9 2 128 729

62. In an experiment, the approximate population P of a bacteria colony is given by, where t is the number of days since start of the experiment. Find the population of the colony on the 8th day. 5. 480 Simplify. All variables represent nonnegative numbers. 6. 7. Lesson Quiz: Part II

63. Lesson Quiz: Part I Multiply. 1. (x + 7) 2 2. (x – 2) 2 3. (5x + 2y) 2 4. (2x – 9y) 2 5. (4x + 5y)(4x – 5y) 6. (m 2 + 2n)(m 2 – 2n) x 2 – 4x + 4 x 2 + 14x + 49 25x 2 + 20xy + 4y 2 4x 2 – 36xy + 81y 2 16x 2 – 25y 2 m 4 – 4n 2

64. Lesson Quiz: Part II 7. Write a polynomial that represents the shaded area of the figure below. 14x – 85 x + 6 x – 6x – 7