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Splash Screen. Ch. 12-3 to 12-7 Simplifying Polynomials. Vocabulary. Monomial is: A number, (e.g. -5, 0, 7, …) A variable, (e.g. a, b, c, ….) or a product of numbers and/ or variables. (e.g. 5*10, 3abc, xyz )

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  1. Splash Screen

  2. Ch. 12-3 to 12-7 Simplifying Polynomials

  3. Vocabulary • Monomial is: • A number, (e.g. -5, 0, 7, …) • A variable, (e.g. a, b, c, ….) • or a product of numbers and/ or variables. (e.g. 5*10, 3abc, xyz ) • Polynomial is an algebraic expression that is the sum or difference of one or more monomials. (e.g. -5a + 7b – 10c + 6a - 5b)

  4. Example 3-1a The like terms in this expression are 3r and –r. Write the polynomial. = 3r – r + 8p – 6q Group like terms Simplify by combining like terms. Answer:

  5. Example 3-1b Answer:

  6. Answer: Example 3-2a There are no like terms in the expression.

  7. Example 3-2b Answer:simplest form

  8. Example 3-3a Rewrite the polynomial from the greatest powers to the least. Thengroup and add like terms. 2x+ 8x²– 9x + 3 – 2x² = 8x² - 2x² + 2x – 9x + 3 = (8 – 2) x² + (2 – 9)x + 3 = 6x² + – 7x + 3 Answer:

  9. Answer: Example 3-3b

  10. Ch. 12-4 Adding Polynomials

  11. Answer: The sum is Example 4-1a Method 1 Add vertically. Align like terms. Add. Method 2 Add horizontally. Associative and Commutative Properties

  12. Answer: Example 4-1b

  13. Ch. 12-5 Subtracting Polynomials

  14. Example 5-1a = 8c + 3 - + 6c - + 2 - + = - = 8c + 3 – 6c - 2 = 8c – 6c + 3 – 2 Combining like terms = (8 - 6)c + (3 – 2) = 2c + 1 The answer is 2c +1

  15. Example 5-1b Answer:

  16. Example 5-3a (6z + 1) – (2z – 5) = 6z + 1 – + 2z - - 5 Remove parenthesis = 6z + 1 – 2z + 5 - + = -; - - = + = 6z – 2z + 1 + 5 Combining like terms = (6 – 2)z + (1 + 5) = 4z + 6) Answer: The difference is 4z + 6

  17. Example 5-3b Answer:

  18. Example 5-4a (10f² - 15) – (- 5f + 3) = 10f² - 15 - - 5f - + 3 = 10f² - 15 + 5f – 3 - - = +, - + = - = 10f² + 5f – 15 – 3 arrange from greatest power to the least = 10f² + 5f -18 Answer: The difference is 10f² + 5f -18

  19. Example 5-4b Answer:

  20. Ch. 12-6 Multiplying and Dividing Monomials

  21. Important Concepts: Product of Powers 3² * 3³ 3² = 3 * 3 3³ = 3 * 3 * 3 So, 3² * 3³ = 3 * 3 * 3 * 3 * 3 = 35 We found out that when multiplying two exponents with the same base, we can simply add their powers. 3² * 3³ = 32+3 = 35 In the same case, r³ * r² = r2+3 = r5

  22. Find . Express using exponents. Example 6-1a The common base is 7. Add the exponents. Check Answer:

  23. Find . Express using exponents. Example 6-1b Answer:

  24. Find . Express using exponents. Example 6-2a Commutative and Associative Properties The common base is x. Add the exponents. Answer:

  25. Find . Express using exponents. Example 6-2b Answer:

  26. Important concepts: Quotient of powers 5³ 5*5*5 5² 5*5 We can also do 5³‾² = 5¹ = 5 In the same way, r r³ This is called Quotient of Powers = 5 = 5 = r ‾³ = r² 5

  27. Find . Express using exponents. Example 6-3a The common base is 6. Simplify. Answer:

  28. Find . Express using exponents. Example 6-3b Answer:

  29. Find . Express using exponents. Example 6-4a The common base is a. Simplify. Answer:

  30. Find . Express using exponents. Example 6-4b Answer:

  31. Example 6-5a UNIT CONVERSION One centimeter is 10 millimeters, and one kilometer is 106 millimeters. How many centimeters are there in one kilometer? To find how many centimeters there are in one kilometer, divide 106 by 10. Quotient of Powers Simplify. Answer: There are 105 centimeters in one kilometer.

  32. Example 6-5b UNIT CONVERSION One decimeter is 10 centimeters, and one kilometer is 105 centimeters. How many decimeters are there in one kilometer? Answer:104 decimeters

  33. End of Lesson 6

  34. Example 7-1a Distributive Property Answer:

  35. Example 7-1b Answer:

  36. Example 7-2a Distributive Property Definition of subtraction Answer:

  37. Example 7-2b Answer:

  38. Example 7-3a Distributive Property Answer:

  39. Example 7-3b Answer:

  40. Example 7-4a Distributive Property Simplify. Definition of subtraction Answer:

  41. Example 7-4b Answer:

  42. Power Rule: (3²)³ = 9³ = 9 * 9 * 9 = 729 (3²)³ = (3²) (3²) (3²) = 9 * 9 * 9 = 729 3²*³ = 3 = 3*3 * 3*3 * 3*3 = 729 We found out that when there is another power over an exponent, we can find the value by multiplying their powers. Same rule applies to algebraic expressions (x³)³ = x³*³ = x 6 9

  43. Example 1: Power of a product (-2xy)³ = -2¹*³x¹*³y¹*³ any # to the 1st power is the number itself = (-2)(-2)(-2) x³y³ -2³ = (-2)(-2)(-2) = -8 =-8x³y³ Answer: -8x³y³

  44. Your turn: (-3ab)² = -3¹*²a¹*²b¹*² any # to the 1st power is the number itself = (-3)(-3) a²b² -3² = (-3)(-3) = 9 = 9a²b² Answer: 9a²b²

  45. Example 2: Power of a power (2²xy³)² = 2²*²x¹*²y³*² any # to the 1st power is the number itself = (2)(2)(2)(2) x²y =16x²y Answer: 4x²y 6 6 6

  46. Your turn: (-3²a³b)² = -3²*²a³*²b¹*² any # to the 1st power is the number itself = -3 a b² = 81a b² Answer: 81a b² 4 6 6 6

  47. Example 3: Product rule and Power rule (4cd)² (-3d²)³ = (4¹*²c¹*²d¹*²) (-3¹*³d²*³) Follow the power rule = (4²c²d²) (-3³d ) = (16c²d²) (-27d ) = (16)(-27)(c²)(d²)(d ) =-432c²d² Product rule =-432c²d 6 6 6 + 6 8

  48. Your turn: 5 6 (-2x )³ (-5xy )² =-200x y 12 17

  49. End of Lesson 7

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