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Splash Screen. Five-Minute Check (over Lesson 10–5) CCSS Then/Now New Vocabulary Example 1: Real-World Example: Use Pascal’s Triangle Key Concept: Binomial Theorem Example 2: Use the Binomial Theorem Example 3: Coefficients Other Than 1 Example 4: Determine a Single Term

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  1. Splash Screen

  2. Five-Minute Check (over Lesson 10–5) CCSS Then/Now New Vocabulary Example 1: Real-World Example: Use Pascal’s Triangle Key Concept: Binomial Theorem Example 2: Use the Binomial Theorem Example 3: Coefficients Other Than 1 Example 4: Determine a Single Term Concept Summary: Binomial Expansion Lesson Menu

  3. Find the first three terms of the sequence.a1 = 2, an + 1 = 3an – 1 A. 2, 5, 14 B. 2, 6, 12 C. 2, 14, 41 D. 2, 5, 8 5-Minute Check 1

  4. Find the first three terms of the sequence.a1 = 2, an + 1 = 3an – 1 A. 2, 5, 14 B. 2, 6, 12 C. 2, 14, 41 D. 2, 5, 8 5-Minute Check 1

  5. Find the first three terms of the sequence.a1 = –1, an + 1 = 5an + 2 A. –1, 0, 7 B. –1, –3, –10 C. –1, –3, –13 D. –3, –8, –13 5-Minute Check 2

  6. Find the first three terms of the sequence.a1 = –1, an + 1 = 5an + 2 A. –1, 0, 7 B. –1, –3, –10 C. –1, –3, –13 D. –3, –8, –13 5-Minute Check 2

  7. Find the first three iterates of the function for the given initial value.f(x) = 4x + 2, x0 = 1 A. 6, 10, 14 B. 6, 26, 106 C. 1, 6, 26 D. 1, 6, 10 5-Minute Check 3

  8. Find the first three iterates of the function for the given initial value.f(x) = 4x + 2, x0 = 1 A. 6, 10, 14 B. 6, 26, 106 C. 1, 6, 26 D. 1, 6, 10 5-Minute Check 3

  9. Find the first three iterates of the function for the given initial value.f(x) = x2 + 1, x0 = 2 A. 5, 26, 677 B. 5, 10, 17 C. 2, 5, 26 D. 2, 10, 17 5-Minute Check 4

  10. Find the first three iterates of the function for the given initial value.f(x) = x2 + 1, x0 = 2 A. 5, 26, 677 B. 5, 10, 17 C. 2, 5, 26 D. 2, 10, 17 5-Minute Check 4

  11. If the rate of inflation is 3%, the cost of an item in future years can be found by iterating the function c(x) = 1.03x. Find the cost of a $15 CD in five years. A. $20.15 B. $18.25 C. $17.39 D. $15.45 5-Minute Check 5

  12. If the rate of inflation is 3%, the cost of an item in future years can be found by iterating the function c(x) = 1.03x. Find the cost of a $15 CD in five years. A. $20.15 B. $18.25 C. $17.39 D. $15.45 5-Minute Check 5

  13. Write a recursive formula for the number of diagonals an of an n-sided polygon. A.an = an – 1 + n – 2 B.an = an – 1 + n C.an = an – 1 + n – 1 D.an = an – 1 + n + 2 5-Minute Check 6

  14. Write a recursive formula for the number of diagonals an of an n-sided polygon. A.an = an – 1 + n – 2 B.an = an – 1 + n C.an = an – 1 + n – 1 D.an = an – 1 + n + 2 5-Minute Check 6

  15. Content Standards A.APR.5 Know and apply the Binomial Theorem for the expansion of (x + y)nin powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined for example by Pascal’s Triangle. Mathematical Practices 4 Model with mathematics. CCSS

  16. You worked with combinations. • Use Pascal’s triangle to expand powers of binomials. • Use the Binomial Theorem to expand powers of binomials. Then/Now

  17. Pascal’s triangle Vocabulary

  18. Use Pascal’s Triangle Expand (p + t)5. Write row 5 of Pascal’s triangle. 1 5 10 10 5 1 Use the patterns of a binomial expansion and thecoefficients to write the expansion of (p + t)5. (p + t)5 = 1p5t0 + 5p4t1 + 10p3t2 + 10p2t3 + 5p1t4 + 1p0t5 = p5 + 5p4t + 10p3t2 + 10p2t3 + 5pt4 + t5 Answer: Example 1

  19. Use Pascal’s Triangle Expand (p + t)5. Write row 5 of Pascal’s triangle. 1 5 10 10 5 1 Use the patterns of a binomial expansion and thecoefficients to write the expansion of (p + t)5. (p + t)5 = 1p5t0 + 5p4t1 + 10p3t2 + 10p2t3 + 5p1t4 + 1p0t5 = p5 + 5p4t + 10p3t2 + 10p2t3 + 5pt4 + t5 Answer:(p + t)5 = p5 + 5p4t + 10p3t2 + 10p2t3 + 5pt4 + t5 Example 1

  20. Expand (x + y)6. A.x6 + 21x5y1 + 35x4y2 + 21x3y3 + 7x2y4 + y6 B. 6x5y + 15x4y2 + 20x3y3 + 15x2y4 + 6xy5 C.x6 – 6x5y + 15x4y2 – 20x3y3 + 15x2y4 – 6xy5 + y6 D.x6 + 6x5y + 15x4y2 + 20x3y3 + 15x2y4 + 6xy5 + y6 Example 1

  21. Expand (x + y)6. A.x6 + 21x5y1 + 35x4y2 + 21x3y3 + 7x2y4 + y6 B. 6x5y + 15x4y2 + 20x3y3 + 15x2y4 + 6xy5 C.x6 – 6x5y + 15x4y2 – 20x3y3 + 15x2y4 – 6xy5 + y6 D.x6 + 6x5y + 15x4y2 + 20x3y3 + 15x2y4 + 6xy5 + y6 Example 1

  22. Concept

  23. Use the Binomial Theorem Expand (t – w)8. Replace n with 8 in the Binomial Theorem. (t – w)8 = t8 + 8C1 t7w + 8C2 t6w2 + 8C3 t5w3 + 8C4 t4w4 + 8C5 t3w5 + 8C6 t2w6 + 8C7 tw7 + w8 Example 2

  24. Use the Binomial Theorem = t8 – 8t7w + 28t6w2 – 56t5w3 + 70t4w4 – 56t3w5 + 28t2w6 – 8tw7 + w8 Answer: Example 2

  25. Use the Binomial Theorem = t8 – 8t7w + 28t6w2 – 56t5w3 + 70t4w4 – 56t3w5 + 28t2w6 – 8tw7 + w8 Answer: (t – w)8=t8 – 8t7w + 28t6w2 – 56t5w3 + 70t4w4 – 56t3w5 + 28t2w6 – 8tw7 + w8 Example 2

  26. Expand (x – y)4. A.x4 + 4x3y + 6x2y2 + 4xy3 + y4 B. 6x3y + 15x2y2 + 20xy3 + 15y4 + 6 C.x4 – 4x3y + 6x2y2 – 4xy3 + y4 D. 4x4 – 4x3y + 6x2y2 – 4xy3 + 4y4 Example 2

  27. Expand (x – y)4. A.x4 + 4x3y + 6x2y2 + 4xy3 + y4 B. 6x3y + 15x2y2 + 20xy3 + 15y4 + 6 C.x4 – 4x3y + 6x2y2 – 4xy3 + y4 D. 4x4 – 4x3y + 6x2y2 – 4xy3 + 4y4 Example 2

  28. Coefficients Other Than 1 Expand (3x – y)4. (3x – y)4 = 4C0(3x)4 + 4C1 (3x)3(–y) + 4C2 (3x)2(–y)2 +4C3 (3x)(–y)3 + 4C4 (–y)4 Answer: Example 3

  29. Coefficients Other Than 1 Expand (3x – y)4. (3x – y)4 = 4C0(3x)4 + 4C1 (3x)3(–y) + 4C2 (3x)2(–y)2 +4C3 (3x)(–y)3 + 4C4 (–y)4 Answer: (3x – y)4 = 81x4 – 108x3y + 54x2y2 – 12xy3 + y4 Example 3

  30. Expand (2x + y)4. A. 16x4 + 32x3y + 24x2y2 + 8xy3 + y4 B. 32x5 + 80x4y + 80x3y2 + 40x2y3 + 10xy4+y5 C. 8x4 + 16x3y + 12x2y + 4xy3 + y4 D. 32x4 + 64x3y + 48x2y2 + 16xy3 + 2y4 Example 3

  31. Expand (2x + y)4. A. 16x4 + 32x3y + 24x2y2 + 8xy3 + y4 B. 32x5 + 80x4y + 80x3y2 + 40x2y3 + 10xy4+y5 C. 8x4 + 16x3y + 12x2y + 4xy3 + y4 D. 32x4 + 64x3y + 48x2y2 + 16xy3 + 2y4 Example 3

  32. Determine a Single Term Find the fourth term in the expansion of (a + 3b)4. First, use the Binomial Theorem to write the expressionin sigma notation. In the fourth term, k = 3. k = 3 Example 4

  33. Determine a Single Term = 108ab3 Simplify. Answer: Example 4

  34. Determine a Single Term = 108ab3 Simplify. Answer: 108ab3 Example 4

  35. Find the fifth term in the expansion of (x + 2y)6. A. 240y4 B. 240x2y4 C. 15x2y4 D. 30x2y4 Example 4

  36. Find the fifth term in the expansion of (x + 2y)6. A. 240y4 B. 240x2y4 C. 15x2y4 D. 30x2y4 Example 4

  37. Concept

  38. End of the Lesson

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