Five-Minute Check (over Lesson 10–5)CCSSThen/NowNew VocabularyExample 1:Real-World Example: Use Pascal’s TriangleKey Concept: Binomial TheoremExample 2:Use the Binomial TheoremExample 3:Coefficients Other Than 1Example 4:Determine a Single TermConcept Summary: Binomial Expansion

Over Lesson 10–5

A. 2, 5, 14

B. 2, 6, 12

C. 2, 14, 41

D. 2, 5, 8

Find the first three terms of the sequence.a1 = 2, an + 1 = 3an – 1

Over Lesson 10–5

A. –1, 0, 7

B. –1, –3, –10

C. –1, –3, –13

D. –3, –8, –13

Find the first three terms of the sequence.a1 = –1, an + 1 = 5an + 2

Over Lesson 10–5

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, x0 = 1

Over Lesson 10–5

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) = x2 + 1, x0 = 2

Over Lesson 10–5

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

A. an = an – 1 + n – 2

B. an = an – 1 + n

C. an = an – 1 + n – 1

D. an = an – 1 + n + 2

Write a recursive formula for the number of diagonals an of an n-sided polygon.

Content StandardsA.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 Practices4 Model with mathematics.

You worked with combinations.

• Use Pascal’s triangle to expand powers of binomials.

• Use the Binomial Theorem to expand powers of binomials.

• Pascal’s triangle

Use Pascal’s Triangle

Expand (p + t)5.Write row 5 of Pascal’s triangle.1 5 10 10 5 1Use the patterns of a binomial expansion and thecoefficients to write the expansion of (p + t)5.

Answer: (p + t)5 = p5 + 5p4t + 10p3t2 + 10p2t3 + 5pt4 + t5

(p + t)5 = 1p5t0 + 5p4t1 + 10p3t2 + 10p2t3 + 5p1t4 + 1p0t5

= p5 + 5p4t + 10p3t2 + 10p2t3 + 5pt4 + t5

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

Expand (x + y)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

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

A. x4 + 4x3y + 6x2y2 + 4xy3 + y4

B. 6x3y + 15x2y2 + 20xy3 + 15y4 + 6

C. x4 – 4x3y + 6x2y2 – 4xy3 + y4

D. 4x4 – 4x3y + 6x2y2 – 4xy3 + 4y4

Expand (x – y)4.

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

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

Expand (2x + y)4.

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

Determine a Single Term

=108ab3

Simplify.

Answer: 108ab3

A. 240y4

B. 240x2y4

C. 15x2y4

D. 30x2y4

Find the fifth term in the expansion of (x + 2y)6.