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- 1. Binomial Theorem
- 2. Binomial TheoremThe expansions of (a + b)n are(a + b)0 = 1
- 3. Binomial TheoremThe expansions of (a + b)n are(a + b)0 = 1(a + b)1 = 1a + 1b
- 4. Binomial TheoremThe expansions of (a + b)n are(a + b)0 = 1(a + b)1 = 1a + 1b(a + b)2 = 1a2 + 2ab + 1b2
- 5. Binomial TheoremThe expansions of (a + b)n are(a + b)0 = 1(a + b)1 = 1a + 1b(a + b)2 = 1a2 + 2ab + 1b2(a + b)3 = 1a3 + 3a2b + 3ab2 + 1b3
- 6. Binomial TheoremThe expansions of (a + b)n are(a + b)0 = 1(a + b)1 = 1a + 1b(a + b)2 = 1a2 + 2ab + 1b2(a + b)3 = 1a3 + 3a2b + 3ab2 + 1b3(a + b)4 = 1a4 + 4a3b + 6a2b2 + 4ab3 + 1b4
- 7. Binomial TheoremThe expansions of (a + b)n are(a + b)0 = 1(a + b)1 = 1a + 1b(a + b)2 = 1a2 + 2ab + 1b2(a + b)3 = 1a3 + 3a2b + 3ab2 + 1b3(a + b)4 = 1a4 + 4a3b + 6a2b2 + 4ab3 + 1b4(a + b)5 = 1a5 + 5a4b + 10a3b2 + 10a2b3 + 5ab4 + 1b5
- 8. Binomial TheoremThe expansions of (a + b)n are(a + b)0 = 1(a + b)1 = 1a + 1b(a + b)2 = 1a2 + 2ab + 1b2(a + b)3 = 1a3 + 3a2b + 3ab2 + 1b3(a + b)4 = 1a4 + 4a3b + 6a2b2 + 4ab3 + 1b4(a + b)5 = 1a5 + 5a4b + 10a3b2 + 10a2b3 + 5ab4 + 1b5We make the following observations about these expansions.
- 9. Binomial TheoremThe expansions of (a + b)n are(a + b)0 = 1(a + b)1 = 1a + 1b(a + b)2 = 1a2 + 2ab + 1b2(a + b)3 = 1a3 + 3a2b + 3ab2 + 1b3(a + b)4 = 1a4 + 4a3b + 6a2b2 + 4ab3 + 1b4(a + b)5 = 1a5 + 5a4b + 10a3b2 + 10a2b3 + 5ab4 + 1b5We make the following observations about these expansions.1. The terms in the expansion may be arranged in order
- 10. Binomial TheoremThe expansions of (a + b)n are(a + b)0 = 1(a + b)1 = 1a + 1b(a + b)2 = 1a2 + 2ab + 1b2(a + b)3 = 1a3 + 3a2b + 3ab2 + 1b3(a + b)4 = 1a4 + 4a3b + 6a2b2 + 4ab3 + 1b4(a + b)5 = 1a5 + 5a4b + 10a3b2 + 10a2b3 + 5ab4 + 1b5We make the following observations about these expansions.1. The terms in the expansion may be arranged in order andthe combined power of a and b of each term is n.
- 11. Binomial TheoremThe expansions of (a + b)n are(a + b)0 = 1(a + b)1 = 1a + 1b(a + b)2 = 1a2 + 2ab + 1b2(a + b)3 = 1a3 + 3a2b + 3ab2 + 1b3(a + b)4 = 1a4 + 4a3b + 6a2b2 + 4ab3 + 1b4(a + b)5 = 1a5 + 5a4b + 10a3b2 + 10a2b3 + 5ab4 + 1b5We make the following observations about these expansions.1. The terms in the expansion may be arranged in order andthe combined power of a and b of each term is n.For example, the terms in the expansion of (a + b)4 are a4,a3b1, a2b2, a1b3, b4.
- 12. Binomial TheoremThe expansions of (a + b)n are(a + b)0 = 1(a + b)1 = 1a + 1b(a + b)2 = 1a2 + 2ab + 1b2(a + b)3 = 1a3 + 3a2b + 3ab2 + 1b3(a + b)4 = 1a4 + 4a3b + 6a2b2 + 4ab3 + 1b4(a + b)5 = 1a5 + 5a4b + 10a3b2 + 10a2b3 + 5ab4 + 1b5We make the following observations about these expansions.1. The terms in the expansion may be arranged in order andthe combined power of a and b of each term is n.For example, the terms in the expansion of (a + b)4 are a4,a3b1, a2b2, a1b3, b4.2. The coefficients all the expansions form the PascalTriangle.
- 13. Binomial TheoremThe Pascal Triangle 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1
- 14. Binomial TheoremThe Pascal Triangle 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1a. The triangle are flanked by 1s to the left and the right.
- 15. Binomial TheoremThe Pascal Triangle 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1a. The triangle are flanked by 1s to the left and the right.b. Each number in the middle is the sum of the two numbersin the row directly above it.
- 16. Binomial TheoremThe Pascal Triangle 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1a. The triangle are flanked by 1s to the left and the right.b. Each number in the middle is the sum of the two numbersin the row directly above it.We may use the Pascal triangle to expand binomials ofsmall power.
- 17. Binomial Theorem 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1Example A. Expand (2x + 3y)3
- 18. Binomial Theorem 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1Example A. Expand (2x + 3y)3From the Pascal triangle(a + b)3 = 1a3 + 3a2b + 3ab2 + 1b3
- 19. Binomial Theorem 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1Example A. Expand (2x + 3y)3From the Pascal triangle(a + b)3 = 1a3 + 3a2b + 3ab2 + 1b3To expand (2x + 3y)3, set (2x) = a, (3y) = b,
- 20. Binomial Theorem 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1Example A. Expand (2x + 3y)3From the Pascal triangle(a + b)3 = 1a3 + 3a2b + 3ab2 + 1b3To expand (2x + 3y)3, set (2x) = a, (3y) = b, we get(2x + 3y)3= 1(2x)3 + 3(2x)2(3y) + 3(2x)(3y)2 + 1(3y)3
- 21. Binomial Theorem 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1Example A. Expand (2x + 3y)3From the Pascal triangle(a + b)3 = 1a3 + 3a2b + 3ab2 + 1b3To expand (2x + 3y)3, set (2x) = a, (3y) = b, we get(2x + 3y)3= 1(2x)3 + 3(2x)2(3y) + 3(2x)(3y)2 + 1(3y)3= 8x3 + 36x2y + 54xy2 + 27y3
- 22. Binomial Theorem 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1Example B. Expand (2x – y)4
- 23. Binomial Theorem 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1Example B. Expand (2x – y)4From the Pascal triangle,(a + b)4 = 1a4 + 4a3b + 6a2b2 + 4ab3 + 1b4
- 24. Binomial Theorem 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1Example B. Expand (2x – y)4From the Pascal triangle,(a + b)4 = 1a4 + 4a3b + 6a2b2 + 4ab3 + 1b4Hence by setting (2x) = a, (-y) = b,
- 25. Binomial Theorem 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1Example B. Expand (2x – y)4From the Pascal triangle,(a + b)4 = 1a4 + 4a3b + 6a2b2 + 4ab3 + 1b4Hence by setting (2x) = a, (-y) = b, we get(2x–y )4= 1(2x)4 + 4(2x)3(-y) + 6(2x)2(-y)2 + 4(2x)(-y)3 + 1(-y)4
- 26. Binomial Theorem 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1Example B. Expand (2x – y)4From the Pascal triangle,(a + b)4 = 1a4 + 4a3b + 6a2b2 + 4ab3 + 1b4Hence by setting (2x) = a, (-y) = b, we get(2x–y )4= 1(2x)4 + 4(2x)3(-y) + 6(2x)2(-y)2 + 4(2x)(-y)3 + 1(-y)4= 16x4 – 32x3y + 24x2y2 – 8xy3 + y4
- 27. Binomial TheoremFor expanding (a + b)n where n is large, the Pascal triangle isnot efficient.
- 28. Binomial TheoremFor expanding (a + b)n where n is large, the Pascal triangle isnot efficient.Instead, we use the following formula for expanding (a + b)n.
- 29. Binomial TheoremFor expanding (a + b)n where n is large, the Pascal triangle isnot efficient.Instead, we use the following formula for expanding (a + b)n.Binomial Theorem
- 30. Binomial TheoremFor expanding (a + b)n where n is large, the Pascal triangle isnot efficient.Instead, we use the following formula for expanding (a + b)n.Binomial TheoremIn the expansion of (a + b)n, the coefficient for the termakbn–k is nCk.
- 31. Binomial TheoremFor expanding (a + b)n where n is large, the Pascal triangle isnot efficient.Instead, we use the following formula for expanding (a + b)n.Binomial TheoremIn the expansion of (a + b)n, the coefficient for the termakbn–k is nCk. n!Reminder: nCk = (n – k)!k!
- 32. Binomial TheoremFor expanding (a + b)n where n is large, the Pascal triangle isnot efficient.Instead, we use the following formula for expanding (a + b)n.Binomial TheoremIn the expansion of (a + b)n, the coefficient for the termakbn–k is nCk. n!Reminder: nCk = (n – k)!k!Example C.In the expansion of (a + b)5, the coefficient for the term a3b2is 5C3.
- 33. Binomial TheoremFor expanding (a + b)n where n is large, the Pascal triangle isnot efficient.Instead, we use the following formula for expanding (a + b)n.Binomial TheoremIn the expansion of (a + b)n, the coefficient for the termakbn–k is nCk. n!Reminder: nCk = (n – k)!k!Example C.In the expansion of (a + b)5, the coefficient for the term a3b2is 5C3.Since 5! = 10, so the term is 10a3b2. (5 – 3)!3!
- 34. Binomial TheoremExample D. Find the exact term with x4y5 in the expansionof (x – 2y)9.
- 35. Binomial TheoremExample D. Find the exact term with x4y5 in the expansionof (x – 2y)9.The a4b5 term in the expansion of (a + b)9 is 9C4a4b5.
- 36. Binomial TheoremExample D. Find the exact term with x4y5 in the expansionof (x – 2y)9.The a4b5 term in the expansion of (a + b)9 is 9C4a4b5.So the x4y5 term in the expansion of (x – 2y)9 is 9C4(x)4(-2y)5.
- 37. Binomial TheoremExample D. Find the exact term with x4y5 in the expansionof (x – 2y)9.The a4b5 term in the expansion of (a + b)9 is 9C4a4b5.So the x4y5 term in the expansion of (x – 2y)9 is 9C4(x)4(-2y)5.Since 9C4 = 9! = 126 (9 – 4)!4!
- 38. Binomial TheoremExample D. Find the exact term with x4y5 in the expansionof (x – 2y)9.The a4b5 term in the expansion of (a + b)9 is 9C4a4b5.So the x4y5 term in the expansion of (x – 2y)9 is 9C4(x)4(-2y)5.Since 9C4 = 9! = 126 (9 – 4)!4!we get that9 C4(x)4(-2y)5 = –126(2)5x4y5
- 39. Binomial TheoremExample D. Find the exact term with x4y5 in the expansionof (x – 2y)9.The a4b5 term in the expansion of (a + b)9 is 9C4a4b5.So the x4y5 term in the expansion of (x – 2y)9 is 9C4(x)4(-2y)5.Since 9C4 = 9! = 126 (9 – 4)!4!we get that9 C4(x)4(-2y)5 = –126(2)5x4y5 = –4032x4y5 HW Find the speific terms in the expansions: 1. the term x7y5 and x9y3 in (x + 2y)12 2. the term x4y5 and x6y3 in (3x + 2y)9 3. the term x10y6 and x13y3 in (3x – y)16 4. the term x10y11 and x13y8 in (2x – 2y)21

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