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Accelerated Math III

Accelerated Math III. Wednesday, September 12, 2012 Binomial Expansion, etc. The average of 5 test scores is 60. After a 6 th test is taken the average of all 6 test scores is now 65. What is the score on the 6 th test?  A) 70 (B) 75 (C) 80 (D) 85 (E) 90.

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Accelerated Math III

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  1. Accelerated Math III Wednesday, September 12, 2012 Binomial Expansion, etc.

  2. The average of 5 test scores is 60. After a 6th test is taken the average of all 6 test scores is now 65. What is the score on the 6th test?  A) 70 (B) 75 (C) 80 (D) 85 (E) 90

  3. Binomial Expansion • What are the terms of (a + b)2? • What are the terms of (a + b)3? • What are the terms of (a + b)4?

  4. Another Look… • (x + y)0 = 1 • (x + y)1 = x + y • (x + y)2 = xx + xy + yx + yy = x2 + 2xy + y2 • (x + y)3 = (x + y)(xx + xy + yx + yy) = xxx + xxy + xyx + xyy + yxx + yxy + yyx + yyy x3 + 3x2y + 3xy2 + y3 • (x + y)4 = (x + y)(xxx + xxy + xyx + xyy + yxx + yxy + yyx + yyy) = xxxx + xxxy + xxyx + xxyy + xyxx + xyxy + xyyx + xyyy + yxxx + yxxy + yxyx + yxyy + yyxx + yyxy + yyyx + yyyy) = x4 + 4x3y + 6x2y2 + 4xy3 + y4

  5. Still another look • (x + y)0 = 1 • (x + y)1 = 1x + 1y • (x + y)2 = 1x2 + 2xy + 1y2 • (x + y)3 = 1x3 + 3x2y + 3xy2 + 1y3 • (x + y)4 = 1x4 + 4x3y + 6x2y2 + 4xy3 + 1y4 • Where have you seen this pattern before?

  6. Generalize… • What are the terms of (2a  b)5? • 32a5 - 80a4b+ 80a3b2 – 40a2b3 + 10ab4 – b5 • Write out the terms of (1 + i )6. • 1 + 6i – 15 – 20i + 15 + 6i – 1 = -8i

  7. Try These! • Write the fourth term of (3  x)7. • C(7, 3) ·34 ·( x)3 = 2835 x3 • Write the third term of (2x + y)9. • C(9, 2) ·(2x)7 ·(y)2 = 4608 x7 y2 • Write the term of (2e )8 that contains e6. • C(8, 2) ·(2e)6 ·()2 = 1792 e6 2

  8. Write the term of (x2 + 1/x)7 that contains x5. • 35x5 • Find the sum of the coefficients of (a + b)5. • 25 = 32 • Find the sum of the coefficients of (a  b)12. • 012 = 0 • Find the sum of the coefficients of (2a + n)4. • 34 = 81

  9. Questions?? Expanding a binomial expression Finding the nth term of a binomial expansion Finding combinations using factorials and/or Pascal’s Triangle Applications…

  10. Homework • Read over pages 734 – 738 in the purple book: • Work problems on Pg. 739: 9, 21, 35, 37, 39, 43, 47, 51, 53, 55, 59, 61, 67, 69, 79, 83, 85, 87

  11. Or… Homework • Read over pages 656 – 660 in the orange book: • Work problems on Pg. 661: 9, 21, 35, 37, 39, 43, 47, 51, 53, 55, 59, 61, 67, 79, 83, 85, 87, 89 - 95

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