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Lesson Menu 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

Over Lesson 10–5 5-Minute Check 1 A.2, 5, 14 B.2, 6, 12 C.2, 14, 41 D.2, 5, 8 Find the first three terms of the sequence. a 1 = 2, a n + 1 = 3a n – 1

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

Over Lesson 10–5 5-Minute Check 3 A.6, 10, 14 B.6, 26, 106 C.1, 6, 26 D.1, 6, 10 Find the first three iterates of the function for the given initial value. f(x) = 4x + 2, x 0 = 1

Over Lesson 10–5 5-Minute Check 4 A.5, 26, 677 B.5, 10, 17 C.2, 5, 26 D.2, 10, 17 Find the first three iterates of the function for the given initial value. f(x) = x 2 + 1, x 0 = 2

Over Lesson 10–5 5-Minute Check 5 A.$20.15 B.$18.25 C.$17.39 D.$15.45 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.

Over Lesson 10–5 5-Minute Check 6 A.a n = a n – 1 + n – 2 B.a n = a n – 1 + n C.a n = a n – 1 + n – 1 D.a n = a n – 1 + n + 2 Write a recursive formula for the number of diagonals a n of an n-sided polygon.

CCSS Content Standards A.APR.5 Know and apply the Binomial Theorem for the expansion of (x + y) n in 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.

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

Vocabulary Pascal’s triangle

Example 1 Use Pascal’s Triangle Expand (p + t) 5. Write row 5 of Pascal’s triangle Use the patterns of a binomial expansion and the coefficients to write the expansion of (p + t) 5. Answer: (p + t) 5 = p 5 + 5p 4 t + 10p 3 t p 2 t 3 + 5pt 4 + t 5 (p + t) 5 =1p 5 t 0 + 5p 4 t p 3 t p 2 t 3 + 5p 1 t 4 + 1p 0 t 5 =p 5 + 5p 4 t + 10p 3 t p 2 t 3 + 5pt 4 + t 5

Example 1 A.x x 5 y x 4 y x 3 y 3 + 7x 2 y 4 + y 6 B.6x 5 y + 15x 4 y x 3 y x 2 y 4 + 6xy 5 C.x 6 – 6x 5 y + 15x 4 y 2 – 20x 3 y x 2 y 4 – 6xy 5 + y 6 D.x 6 + 6x 5 y + 15x 4 y x 3 y x 2 y 4 + 6xy 5 + y 6 Expand (x + y) 6.

Concept

Example 2 Use the Binomial Theorem Expand (t – w) 8. Replace n with 8 in the Binomial Theorem. (t – w) 8 =t C 1 t 7 w + 8 C 2 t 6 w C 3 t 5 w C 4 t 4 w C 5 t 3 w C 6 t 2 w C 7 tw 7 + w 8

Example 2 Use the Binomial Theorem =t 8 – 8t 7 w + 28t 6 w 2 – 56t 5 w t 4 w 4 – 56t 3 w t 2 w 6 – 8tw 7 + w 8 Answer: (t – w) 8 = t 8 – 8t 7 w + 28t 6 w 2 – 56t 5 w t 4 w 4 – 56t 3 w t 2 w 6 – 8tw 7 + w 8

Example 2 A.x 4 + 4x 3 y + 6x 2 y 2 + 4xy 3 + y 4 B.6x 3 y + 15x 2 y xy y C.x 4 – 4x 3 y + 6x 2 y 2 – 4xy 3 + y 4 D.4x 4 – 4x 3 y + 6x 2 y 2 – 4xy 3 + 4y 4 Expand (x – y) 4.

Example 3 Coefficients Other Than 1 Expand (3x – y) 4. (3x – y) 4 = 4 C 0 (3x) C 1 (3x) 3 (–y) + 4 C 2 (3x) 2 (–y) C 3 (3x)(–y) C 4 (–y) 4 Answer: (3x – y) 4 = 81x 4 – 108x 3 y + 54x 2 y 2 – 12xy 3 + y 4

Example 3 A.16x x 3 y + 24x 2 y 2 + 8xy 3 + y 4 B.32x x 4 y + 80x 3 y x 2 y xy 4 + y 5 C.8x x 3 y + 12x 2 y + 4xy 3 + y 4 D.32x x 3 y + 48x 2 y xy 3 + 2y 4 Expand (2x + y) 4.

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

Example 4 Determine a Single Term =108ab 3 Simplify. Answer: 108ab 3

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

Concept

End of the Lesson