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Unit 6: Polynomials 6. 1 Objectives The student will be able to:

Unit 6: Polynomials 6. 1 Objectives The student will be able to:. 1. multiply monomials. simplify expressions with monomials. Hernandez – Henry Ford High School. A monomial is a. 1. number, 2. variable, or 3. a product of one or more numbers and variables. Examples: 5 y 3x 2 y 3.

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Unit 6: Polynomials 6. 1 Objectives The student will be able to:

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  1. Unit 6: Polynomials6. 1 ObjectivesThe student will be able to: • 1. multiply monomials. • simplify expressions with monomials. Hernandez – Henry Ford High School

  2. A monomial is a 1. number, 2. variable, or 3. a product of one or more numbers and variables. Examples: 5 y 3x2y3

  3. Why are the following not monomials?x + y addition division 2 - 3a subtraction

  4. Multiplying Monomials When multiplying monomials, you ADD the exponents. 1) x2 • x4 x2+4 x6 2) 2a2y3 • 3a3y4 6a5y7

  5. Simplify m3(m4)(m) • m7 • m8 • m12 • m13

  6. Power of a Power When you have an exponent with an exponent, you multiply those exponents. 1) (x2)3 x2• 3 x6 2) (y3)4 y12

  7. Simplify (p2)4 • p2 • p4 • p8 • p16

  8. Power of a Product When you have a power outside of the parentheses, everything in the parentheses is raised to that power. 1) (2a)3 23a3 8a3 2) (3x)2 9x2

  9. Simplify (4r)3 • 12r3 • 12r4 • 64r3 • 64r4

  10. Power of a Monomial This is a combination of all of the other rules. 1) (x3y2)4 x3• 4 y2• 4 x12 y8 2) (4x4y3)3 64x12y9

  11. Simplify (3a2b3)4 • 12a8b12 • 81a6b7 • 81a16b81 • 81a8b12

  12. 6.2 ObjectivesThe student will be able to: • 1. divide monomials. • simplify negative exponents.

  13. Dividing Monomials When dividing monomials, subtract the exponents. 1. 2. = m6 n3 = b5-2 = b3 = m7-1n5-2

  14. = xy = 9a3b2

  15. Simplify • 48g2h2 • 48gh2 • 4g2h2 • 4gh2

  16. Here’s a tricky one! = 1m0n = n What happened to the m? They canceled out! There are no m’s left over! This leads us to our next rule…

  17. Zero Exponents Anything to the 0 power is equal to 1. a0 = 1 True or False? Anything divided by itself equals one. True! See for yourself!

  18. Negative Exponents A negative exponent means you move the base to the other side of the fraction and make the exponent positive. Notice that the base with the negative exponent moved and became positive!

  19. Simplify. • x-4 y0You can not have negative or zero exponents in your answer.

  20. Simplify • p2 • p12 • . • .

  21. Simplify. You can’t leave the negative exponent! There is another way of doing this without negative exponents. If you don’t want to see it, skip the next slide!!!

  22. Simplify (alternate version). Look and see (visualize) where you have the larger exponent and leave the variable in that location. Subtract the smaller exponent from the larger one. In this problem, r is larger in the numerator and s is larger in the denominator. Notice that you did not have to work with negative exponents! This method is quicker!

  23. Simplify. Get rid of the negative exponent.

  24. Simplify. Get rid of the negative exponents.

  25. Simplify. Get rid of the parentheses. Get rid of the negative exponents.

  26. Simplify • . • . • . • .

  27. 6.3 ObjectiveThe student will be able to: express numbers in scientific and decimal notation.

  28. How wide is our universe? 210,000,000,000,000,000,000,000 miles (22 zeros) This number is written in decimal notation. When numbers get this large, it is easier to write them in scientific notation.

  29. Scientific Notation A number is expressed in scientific notation when it is in the form a x 10n where a is between 1 and 10 and n is an integer

  30. Write the width of the universe in scientific notation. 210,000,000,000,000,000,000,000 miles Where is the decimal point now? After the last zero. Where would you put the decimal to make this number be between 1 and 10? Between the 2 and the 1

  31. 2.10,000,000,000,000,000,000,000. How many decimal places did you move the decimal? 23 When the original number is more than 1, the exponent is positive. The answer in scientific notation is 2.1 x 1023

  32. 1) Express 0.0000000902 in scientific notation. Where would the decimal go to make the number be between 1 and 10? 9.02 The decimal was moved how many places? 8 When the original number is less than 1, the exponent is negative. 9.02 x 10-8

  33. Write 28750.9 in scientific notation. • 2.87509 x 10-5 • 2.87509 x 10-4 • 2.87509 x 104 • 2.87509 x 105

  34. 2) Express 1.8 x 10-4 in decimal notation. 0.00018 3) Express 4.58 x 106 in decimal notation. 4,580,000 On the graphing calculator, scientific notation is done with the button. 4.58 x 106 is typed 4.58 6

  35. 4) Use a calculator to evaluate: 4.5 x 10-5 1.6 x 10-2 Type 4.5 -5 1.6 -2 You must include parentheses if you don’t use those buttons!! (4.5 x 10 -5) (1.6 x 10 -2) 0.0028125 Write in scientific notation. 2.8125 x 10-3

  36. 5) Use a calculator to evaluate: 7.2 x 10-9 1.2 x 102On the calculator, the answer is: 6.E -11 The answer in scientific notation is 6 x 10 -11 The answer in decimal notation is 0.00000000006

  37. 6) Use a calculator to evaluate (0.0042)(330,000).On the calculator, the answer is 1386. The answer in decimal notation is 1386 The answer in scientific notation is 1.386 x 103

  38. 7) Use a calculator to evaluate (3,600,000,000)(23).On the calculator, the answer is: 8.28 E +10 The answer in scientific notation is 8.28 x 10 10 The answer in decimal notation is 82,800,000,000

  39. Write (2.8 x 103)(5.1 x 10-7) in scientific notation. • 14.28 x 10-4 • 1.428 x 10-3 • 14.28 x 1010 • 1.428 x 1011

  40. Write in PROPER scientific notation.(Notice the number is not between 1 and 10) 8) 234.6 x 109 2.346 x 1011 9) 0.0642 x 104 on calculator: 642 6.42 x 10 2

  41. Write 531.42 x 105 in scientific notation. • .53142 x 102 • 5.3142 x 103 • 53.142 x 104 • 531.42 x 105 • 53.142 x 106 • 5.3142 x 107 • .53142 x 108

  42. 6. 4 ObjectivesThe student will be able to: 1. find the degree of a polynomial. 2. arrange the terms of a polynomial in ascending or descending order. SOL: none Designed by Skip Tyler, Varina High School

  43. What does each prefix mean? mono one bi two tri three

  44. What about poly? one or more A polynomial is a monomial or a sum/difference of monomials. Important Note!! An expression is not a polynomial if there is a variable in the denominator.

  45. State whether each expression is a polynomial. If it is, identify it. 1) 7y - 3x + 4 trinomial 2) 10x3yz2 monomial 3) not a polynomial

  46. Which polynomial is represented by X 1 X 1 X2 X • x2 + x + 1 • x2 + x + 2 • x2 + 2x + 2 • x2 + 3x + 2 • I’ve got no idea!

  47. The degree of a monomial is the sum of the exponents of the variables.Find the degree of each monomial. 1) 5x2 2 4a4b3c 8 -3 0

  48. To find the degree of a polynomial, find the largest degree of the terms. 1) 8x2 - 2x + 7 Degrees: 2 1 0 Which is biggest? 2 is the degree! 2) y7 + 6y4 + 3x4m4 Degrees: 7 4 8 8 is the degree!

  49. Find the degree of x5 – x3y2 + 4 • 0 • 2 • 3 • 5 • 10

  50. A polynomial is normally put in ascending or descending order. What is ascending order? Going from small to big exponents. What is descending order? Going from big to small exponents.

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