1. Algebraic Operations

2. Goals Add and subtract algebraic expressions and simplify like terms by applying commutative, associative, and distributive properties. Identify monomial, binomial, and polynomial terms in an algebraic expression. Big Idea Algebraic expressions are often a sum of terms. An expression can be simplified by combining like terms.

3. Algebraic Operations Goal Identify monomial, binomial, and polynomial terms in an algebraic expression. Describe the order of the terms in a polynomial as ascending or descending. Big Idea A polynomial is an algebraic expression involving a sum of terms. Polynomial with 1, 2 or 3 terms are given special names: monomial - 1 term e.g. 5x or 5 or 3abc binomial - 2 terms e.g. 1 + x or 3x 2 + 2 trinomial - 3 terms e.g. 1 + x + x2 or ab + bc + 3

4. Algebraic Operations Goal Add and subtract algebraic expressions and simplify like terms by applying commutative, associative, and distributive properties. Big Idea A polynomial can be simplified by combining like terms. Like terms have variables with the same power. Remember x 0 = 1

5. Algebraic Operations HW Due Tuesday 11/23 176.2, and any 4 of 176.3-26, 181.5, any 3 of 181.19-29, and any 4 of 181.30-47, 181.48, 181.51, any 5 of 181.52-73

6. Algebraic Operations Goal Express the product of terms with the same base using a sum of exponents. Big Idea x a x b = x a+b 2 2 2 = 2 3 ab 2 a = a 2 b 2 For example:

7. Algebraic Operations x a x b = x a+b 3 2 3 = x 2 x = x 2 y x y = x 2 y x y 2 = Apply the rule:

8. Algebraic Operations Goal Express the power of a power using a product of exponents. Big Idea (x a ) b = x ab 2 2 2 = 2 3 ab 2 a 3 = a 4 b 2 Compare with: (2 2 ) 3 = 2 6 (ab 2 ) 3 = a 3 b 6 For example:

9. Algebraic Operations (x a ) b = x ab (3 2 ) 3 = (xy 2 ) 2 = x (xy) 2 = xy 2 (xy) 3 = Apply the rule:

10. HW Due Tuesday 11/23 176.2, and any 4 of 176.3-26 HW Due Wednesday 11/24 181.5, any 3 of 181.19-29, and any 4 of 181.30-47, 181.48, 181.51, any 5 of 181.52-73

11. Algebraic Operations Goal Add and subtract algebraic expressions and simplify like terms by applying commutative, associative, and distributive properties. Big Idea Monomials can be multiplied: Group variables and numerical factors Multiply the numerical factors Multiply the variables For example: 6x 2 (-3x 4 )=6(-3)x 2 x 4 =-18 x 6

12. Algebraic Operations Big Idea Monomials can be multiplied: Group variables and numerical factors Multiply the numerical factors Multiply the variables 3x 2 2x 4 = x 2 y2x= x 3 y xy=

13. Algebraic Operations Goal Add and subtract algebraic expressions and simplify like terms by applying commutative, associative, and distributive properties. Big Idea Binomials can be multiplied by monomials: Group variables and numerical factors Distribute the monomial over the sum Multiply the numerical factors Multiply the variables For example: 6x 2 (-3x 4 +2)=6(-3)x 2 x 4 + 6x 2 (2)=-18 x 6 + 12x 2

14. Algebraic Operations Big Idea Binomials can be multiplied by monomials: Group variables and numerical factors Distribute the monomial over the sum Multiply the numerical factors Multiply the variables x 2 (2+x)= ab (3a + b) = x 2 y (3xy + 2) =

15. Algebraic Operations Goal Add and subtract algebraic expressions and simplify like terms by applying commutative, associative, and distributive properties. HW Due Wednesday 11/24 181.5, any 3 of 181.19-29, and any 4 of 181.30-47, 181.48, 181.51, any 5 of 181.52-73

16. Algebraic Operations HW Due Monday 11/29 185.2, 185.21, 185.22, 185.24, 185.25, 185.28, 185.29, 185.32, 185.35, 185.36 and 185.43 Goal Add and subtract algebraic expressions and simplify like terms by applying commutative, associative, and distributive properties.

17. Algebraic Operations Big Idea Polynomials can be multiplied by monomials: Group variables and numerical factors Distribute the monomial over the sum Multiply the numerical factors Multiply the variables For example: (6x 2 ) (-3x 4 +2)= 6(-3)x 2 x 4 + 6x 2 (2)=-18 x 6 + 12x 2

18. Try these:

21. Algebraic Operations Big Idea Polynomials can be multiplied by polynomials: Group variables and numerical factors Distribute the monomial over the sum Multiply the numerical factors Multiply the variables For example: (6x 2 -1) (-3x 4 +2)= 6(-3)x 2 x 4 + 6x 2 (2) + (-1)(-3)x 4 + (-1)(2) =-18 x 6 + 12x 2 + 3x 4 - 2

22. Algebraic Operations Big Idea Polynomials can be multiplied by polynomials: Group variables and numerical factors Distribute the monomial over the sum Multiply the numerical factors Multiply the variables (5a -1) (3a + 2) = (-1 + x) (2x 2 + 1) = (a + b) (a + b) =

23. Algebraic Operations HW Due Monday 11/29 185.2, 185.21, 185.22, 185.24, 185.25, 185.28, 185.29, 185.32, 185.35, 185.36 and 185.43 Goal Add and subtract algebraic expressions and simplify like terms by applying commutative, associative, and distributive properties.

24. Algebraic Operations Big Idea Polynomials can be multiplied by polynomials: Group variables and numerical factors Distribute the monomial over the sum Multiply the numerical factors Multiply the variables

25. Algebraic Operations What are the four terms if …?

26. Algebraic Operations What are the four terms if …?

27. Algebraic Operations What are the four terms if …?

28. Try these:

30. Algebraic Operations Goal Add and subtract algebraic expressions and simplify like terms by applying commutative, associative, and distributive properties. Try this

31. Algebraic Operations Goal Add and subtract algebraic expressions and simplify like terms by applying commutative, associative, and distributive properties. Try this

32. Algebraic Operations Goal Add and subtract algebraic expressions and simplify like terms by applying commutative, associative, and distributive properties. So what is ?

33. Algebraic Operations Big Idea Mathematical methods often work backwards from the answer to the question. Goal Add and subtract algebraic expressions and simplify like terms by applying commutative, associative, and distributive properties. Which of these expressions is equivalent to 121 – x 2 ?

34. Algebraic Operations If this is the answer, what is the question? Which of these expressions is equivalent to 9x 2 – 16?

35. Algebraic Operations If this is the answer, what is the question? Which of these expressions is equivalent to 9x 2 – 100?

36. Algebraic Operations If this is the answer, what is the question? Which of these expressions is equivalent to 2x 2 + 10x - 12?

37. Algebraic Operations If this is the answer, what is the question? Which of these expressions is equivalent to 3x 2 - 3x - 18?

38. Algebraic Operations HW Due Monday 11/29 185.2, 185.21, 185.22, 185.24, 185.25, 185.28, 185.29, 185.32, 185.35, 185.36 and 185.43 Goal Add and subtract algebraic expressions and simplify like terms by applying commutative, associative, and distributive properties. HW Due Tuesday 11/30 185.44, 186.46, 187.1, 187.2, any 3 of 187.3-18, 188.20, 188.22, any 3 of 188.25-32, 191.7, 191.15, and 191.

39. Algebraic Operations Goal Express and manipulate numbers using scientific notation. Add and subtract algebraic expressions and simplify like terms by applying commutative, associative, and distributive properties. HW Due Wednesday 12/1 196.8, 196.20, 196.27, 196.28, 196.38, 196.39, 196. 45, 196. 47, 196.48, 199.2, any 4 of 199.3-26, and 199.28

40. Algebraic Operations Goal Add and subtract algebraic expressions and simplify like terms by applying commutative, associative, and distributive properties. Big Idea For example

41. Algebraic Operations Big Idea Do these

42. Algebraic Operations Goal Express and manipulate numbers using scientific notation. Big Idea Powers of 10 can be checked by counting shifts of the decimal point that give 1.0

43. Algebraic Operations Big Idea In simplest form the base should have a non-zero digit in the “ones place.”

44. Algebraic Operations Goal Express and manipulate numbers using scientific notation. Write these numbers in simplest form using scientific notation:

45. Algebraic Operations Big Idea Scientific notation simplifies calculations Examples

46. Algebraic Operations The quotient of (9.2x10 6 ) and (2.3x10 2 ) expressed in scientific notation is _______________. What is the product of 12 and 4.2x10 6 expressed in scientific notation? Big Idea Scientific notation simplifies calculations

47. Algebraic Operations Do these with and without the calculator

49. Big Idea The multiplicative inverse of a monomial is 1 divided by the monomial. Algebraic Operations For example:

50. Algebraic Operations Big Idea Polynomials can be divided by monomials: Group variables and numerical factors Distribute the monomial over the sum Multiply the numerical factors Multiply the variables For example:

51. Algebraic Operations Try these:

52. Algebraic Operations Try these (where x, y, z, p, a, and b ≠ 0):

53. Algebraic Operations Try these (where y and a ≠ 0):

54. Algebraic Operations Goal Express and manipulate numbers using scientific notation. Add and subtract algebraic expressions and simplify like terms by applying commutative, associative, and distributive properties. HW Due Wednesday 12/1 196.8, 196.20, 196.27, 196.28, 196.38, 196.39, 196.45, 196. 47, 196.48, 199.2, any 4 of 199.3-26, and 199.28

55. Algebraic Operations Goal Add and subtract algebraic expressions and simplify like terms by applying commutative, associative, and distributive properties. Homework Due Thursday 12/2 201.1, 201.2, 201.3 and Test Review 6

56. Algebraic Operations Big Idea The same algorithm used to divide one number by another number can be used to divide a polynomial by a binomial.

57. Algebraic Operations Big Idea The same algorithm used to divide one number by another number can be used to divide a polynomial by a binomial. To check multiply each term by

58. Algebraic Operations If this is the answer, what is the question? Which of these expressions is equivalent to 3x 2 - 3x - 18? Express the division of a trinomial by two different binomials using this result:

59. Algebraic Operations Goal Add and subtract algebraic expressions and simplify like terms by applying commutative, associative, and distributive properties. Homework Due Thursday 12/2 201.1, 201.2, 201.3, and Test Review 6 Work 201.3 by using “if this is the answer, what is the question?” thinking rather than the long division method described on page 200.