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Splash Screen. Pascal’s triangle. P. 699. Vocabulary. 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 the coefficients to write the expansion of ( p + t ) 5.

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  1. Splash Screen

  2. Pascal’s triangle P. 699 Vocabulary

  3. 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

  4. 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

  5. P. 700 Concept

  6. 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

  7. 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

  8. 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

  9. 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

  10. 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

  11. 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

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

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

  14. Concept

  15. Homework P. 702 # 15 – 31 odd (Show work)

  16. End of the Lesson

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