Warm-up Time-limit….. 5 P 2 = 11 ! = 7 C 3 = 7!___ = 3!(7-3)! 5 C 2 + 5 C 3 =

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Presentation transcript:

Warm-up Time-limit….. 5 P 2 = 11 ! = 7 C 3 = 7!___ = 3!(7-3)! 5 C C 3 =

Warm-up Time-limit….. 5 P 2 = ! = 39,916,800 7 C 3 = 35 7!___ = 35 3!(7-3)! 5 C C 3 = 20

Pascal’s Triangle

Don’t open your book!!! Follow the diagram on the next slide. Answer the questions about the diagram. This is a timed exercise. You have 15 minutes. Work with your group. Each group will hand in one worksheet. 20 points per person. Make sure all names are on the worksheet.

Pascal’s Triangle What would the values be in the empty row? 2. Is there a pattern found in the sums of the rows? 3. What is the pattern of the yellow diagonal row? 4.Can you find the powers of 11? Example 11 0 = 1, 11 1 = 11 etc. Extra Credit (25 pts). Find 2 more patterns…. draw pascal’s diagram and show the describe the patterns. Due by Friday 1/30/09.

2. Powers of = = = =8 2 4 = = = 64 Pascal’s Triangle What would the values be in the empty row? 2. Is there a pattern found in the sums of the rows? 3. What is the pattern of the yellow diagonal row? 4.Can you find the powers of 11? Example 11 0 = 1, 11 1 = 11 etc. Extra Credit (25 pts). Find 2 more patterns…. draw pascal’s diagram and show the describe the patterns. Due by Friday 1/30/ Natural numbers. 4.Look at each row. Row 3… Row 4 …

Expand this Binomial (a + b) 3

Using Pascal’s Triangle to solve (a + b) 3 1.Use the row that has a “3” as the second number Now expand (a + b). 1a b + 3a b + 3 a b + 1a b 3. Insert the exponents (exponents for a start at 3 ….0) 1a 3 b 0 + 3a 2 b a 1 b 2 + 1a 0 b 3 (exponents for b start 0 … 3) 4. Now simplify. a 3 + 3a 2 b + 3ab 2 + b 3

What about (x -2 ) 4 What row in Pascal’s Triangle would you look at? Remember to expand (a + b), let a = x and b = -2 So you will get… ab + 4ab + 6ab + 4ab + 1ab Now insert the exponents. a (4 - 0) b ( 0 4) 1a 4 + 4a 3 b + 6a 2 b 2 + 4ab 3 + 1b 4 Next insert (-2) for all b’s 1a 4 + 4a 3 (-2) + 6a 2 (-2) 2 + 4a(-2) 3 + 1(-2) 4

What about (x -2 ) 4 Finally, simplify this expression 1a 4 + 4a 3 (-2) + 6a 2 (-2) 2 + 4a(-2) 3 + 1(-2) 4 The result will be… a 4 - 8a a a + 16 HOMEWORK Page evens