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
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 = –1, an + 1 = 5an + 2
B. –1, –3, –10
C. –1, –3, –13
D. –3, –8, –13
5-Minute Check 2

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

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

7. 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

8. 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

9. Mathematical Practices 4 Model with mathematics.
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.
CCSS

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

11. Pascal’s triangle
Vocabulary

12. Write row 5 of Pascal’s triangle. 1 5 10 10 5 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.
(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

13. 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

14. Concept

15. Replace n with 8 in the Binomial Theorem.
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 t4w C5 t3w5 + 8C6 t2w6 + 8C7 tw7 + w8
Example 2

16. = t8 – 8t7w + 28t6w2 – 56t5w3 + 70t4w4 – 56t3w5 + 28t2w6 – 8tw7 + w8
Use the Binomial Theorem
= t8 – 8t7w + 28t6w2 – 56t5w3 + 70t4w4 – 56t3w t2w6 – 8tw7 + w8
Answer: (t – w)8 = t8 – 8t7w + 28t6w2 – 56t5w3 + 70t4w4 – 56t3w5 + 28t2w6 – 8tw7 + w8
Example 2

17. 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

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

19. 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

20. Find the fourth term in the expansion of (a + 3b)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

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

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

23. Concept

24. End of the Lesson