1. LESSON 10–5
The Binomial Theorem

2. Five-Minute Check (over Lesson 10-4) TEKS Then/Now New Vocabulary
Example 1: Power of a Binomial Sum
Example 2: Power of a Binomial Difference
Key Concept: Formula for the Binomial Coefficients of (a + b)n
Example 3: Find Binomial Coefficients
Example 4: Binomials with Coefficients Other than 1
Example 5: Real-World Example: Use Binomial Coefficients
Key Concept: Binomial Theorem
Example 6: Expand a Binomial Using the Binomial Theorem
Example 7: Write a Binomial Expansion Using Sigma Notation
Lesson Menu

3. Prove that 3 + 4 + 5 + … + (n + 2) = is valid for all positive integers n.
5–Minute Check 1

4. A.
5–Minute Check 1

5. B.
5–Minute Check 1

6. C.
5–Minute Check 1

7. D.
5–Minute Check 1

8. Prove that 3 + 4 + 5 + … + (n + 2) = is valid for all positive integers n.B. C. D.
5–Minute Check 1

9. Which of the following is an expression that is divisible by 3 for all positive integers n?
B. 3n – 1
C. 4n – 1
D. 5n – 1
5–Minute Check 2

10. Targeted TEKS
P.5(F) Apply the Binomial
Theorem for the expansion of (a + b)n in
powers of a and b for a positive integer n, where
a and b are any numbers.
Mathematical Processes
P.1(A), P.1(G)

11. You represented infinite series using sigma notation. (Lesson 10-1)
Use Pascal’s Triangle to write binomial expansions.
Use the Binomial Theorem to write and find the coefficients of specified terms in binomial expansions.
Then/Now

12. binomial coefficients Pascal’s triangle Binomial Theorem
Vocabulary

13. A. Use Pascal’s Triangle to expand (a + b)6.
Power of a Binomial Sum
A. Use Pascal’s Triangle to expand (a + b)6.
Step 1 Write the series for (a + b)6, omitting the coefficients. Because the power is 6, this series should have or 7 terms. Use the pattern of increasing and decreasing exponents to complete the series.
a6b0 + a5b1 + a4b2 + a3b3 + a2b4 + a1b5 + a0b6 Exponents of a decrease from 6 to 0.
Exponents of b increase from 0 to 6.
Example 1

14. (a + b)6 = 1a6b0 + 6a5b1 + 15a4b2+ 20a3b3 + 15a2b4 + 6a1b5 + 1a0b6
Power of a Binomial Sum
Step 2 Use the numbers in the sixth row of Pascal’s triangle as the coefficients of the terms. To find these numbers, extend Pascal’s triangle to the 6th row by adding corresponding numbers in the previous row.
th row
th row
(a + b)6 = 1a6b0 + 6a5b1 + 15a4b2+ 20a3b3 + 15a2b a1b5 + 1a0b6
(a + b)6 = a6 + 6a5b + 15a4b2+ 20a3b3 + 15a2b ab5 + b6 Simplify.
Example 1

15. Answer: a6 + 6a5b + 15a4b2+ 20a3b3 + 15a2b4 + 6ab5 + b6
Power of a Binomial Sum
Answer: a6 + 6a5b + 15a4b2+ 20a3b3 + 15a2b4 + 6ab5 + b6
Example 1

16. B. Use Pascal’s Triangle to expand (3x + 2)5.
Power of a Binomial Sum
B. Use Pascal’s Triangle to expand (3x + 2)5.
Step 1 Write the series for (a + b)5, omitting the coefficients and replacing a with 3x and b with 2. The series has or 6 terms.
(3x)5(2)0 + (3x)4(2)1 + (3x)3(2)2 + (3x)2(2)3 + (3x)1(2)4 + (3x)0(2)5 Exponents of 3x decrease from 5 to 0.
Exponents of 2 increase from 0 to 5.
Example 1

17. Power of a Binomial Sum
Step 2 The numbers in the 5th row of Pascal’s triangle are 1, 5, 10, 10, 5, and 1. Use these numbers as the coefficients of the terms in the series. Then simplify.
(3x + 2)5 = 1(3x)5(2)0 + 5(3x)4(2)1 + 10(3x)3(2) (3x)2(2)3 + 5(3x)1(2)4 + 1(3x)0(2)5
(3x + 2)5 = 243x x x x x + 32
Answer: 243x x x x x + 32
Example 1

18. Use Pascal’s Triangle to expand (2x + 3)6.
A. 64x x x x x x + 729
B. 64x6 + 96x x x x x + 729
C. 64x x x x x x + 729
D. 32x5 + 48x4 + 72x x x + 243
Example 1

19. Use Pascal’s Triangle to expand (x – 2y)6.
Power of a Binomial Difference
Use Pascal’s Triangle to expand (x – 2y)6.
Because (x – 2y)6 = [x + (–2y)]6 , write the series for (a + b)6 , replacing a with x and b with –2y.
Use the numbers in the 6th row of Pascal’s triangle, 1, 6, 15, 20, 15, 6, and 1, as the binomial coefficients. Then simplify.
(x – 2y)6 = 1x6(–2y)0 + 6x5(–2y)1 + 15x4(–2y) x3(–2y)3 + 15x2(–2y)4 + 6x1(–2y) x0(–2y)6
= x6 – 12x5y + 60x4y2 – 160x3y x2y4 – 192xy5 + 64y6
Example 2

20. Answer: x6 – 12x5y + 60x4y2 – 160x3y3 + 240x2y4 – 192xy5 + 64y6
Power of a Binomial Difference
Answer: x6 – 12x5y + 60x4y2 – 160x3y x2y4 – 192xy5 + 64y6
Example 2

21. Use Pascal’s Triangle to expand (3x – y)6.
A. 729x6 – 243x5y + 81x4y2 – 27x3y3 + 9x2y4 – 3xy y6
B. 729x6 – 1458x5y x4y2 – 540x3y x2y4 – 18xy5 + y6
C. 729x x5y x4y x3y x2y xy5 + y6
D. 3x6 – 18x5y + 45x4y2 – 60x3y3 + 45x2y4 – 18xy5 + y6
Example 2

22. Key Concept 3

23. Find the coefficient of the fourth term in the expansion of (a – b)10.
Find Binomial Coefficients
Find the coefficient of the fourth term in the expansion of (a – b)10.
To find the coefficient of the fourth term, evaluate nCr for n = 10 and r = 4 – 1 or 3.
Subtract.
Rewrite 10! as 10 • 9 • 8 • 7! and divide out common factorials.
Example 3

24. Find Binomial Coefficients
Simplify.
The coefficient of the fourth term in the expansion of (a – b)10 is 120.
Answer: 120
Example 3

25. Find the coefficient of the 5th term in the expansion of (a – b)8.
Example 3

26. Find the coefficient of the x7y term in the expansion of (3x – 4y)8.
Binomials with Coefficients Other than 1
Find the coefficient of the x7y term in the expansion of (3x – 4y)8.
For (3x – 4y)8 to have the form (a + b)n, let a = 3x and b = –4y. The coefficient of the term containing an – r br in the expansion of (a + b)n is given by nCr . So, to find the binomial coefficient of the term containing a7b in the expansion of (a + b)8, evaluate nCr for n = 8 and r = 1.
Subtract.
Example 4

27. Rewrite 8! as 8 • 7! and divide out common factorials.
Binomials with Coefficients Other than 1
Rewrite 8! as 8 • 7! and divide out common factorials.
Simplify.
Thus, the binomial coefficient of the a7b term in (a + b)8 is 8. Substitute 3x for a and –4y for b to find the coefficient of the x7y term in the original binomial expansion.
8a7b = 8(3x)7(–4y) a = 3x and b = –4y
= –69,984x7y Simplify.
Therefore, the coefficient of the x7y term in the expansion of (3x – 4y)8 is –69,984.
Example 4

28. Binomials with Coefficients Other than 1
Answer: –69,984
Example 4

29. Find the coefficient of the x 2y 4 term in the binomial expansion of (2x – 3y)6.
Example 4

30. Use Binomial Coefficients
GAMES During a player’s turn in a certain board game, players must spin a spinner. The four possible colors the spinner can land on are green, blue, red, or yellow. If the probability for all four colors is equal, what is the probability of landing on green 5 times out of 10 spins?
A success in this situation is landing on green, so p = and q = 1 – or .
Example 5

31. Use Binomial Coefficients
Each spin represents a trial, so n = 10. You want to find the probability that the player lands on green 5 times out of those 10 trials, so let x = 5. To find this probability, find the value of the term nCx px qn – x in the expansion of (p + q)n .
p = , q = , n = 10, and x = 5
Use a calculator.
Example 5

32. Use Binomial Coefficients
So, the probability of landing on green 5 times out of 10 spins is about 5.84%.
Answer: about 5.84%
Example 5

33. COIN TOSS A fair coin is flipped 10 times
COIN TOSS A fair coin is flipped 10 times. Find the probability of getting exactly 3 tails. A. B. C. D.
Example 5

34. Key Concept 6

35. A. Use the Binomial Theorem to expand (2t + 3u)3.
Expand a Binomial Using the Binomial Theorem
A. Use the Binomial Theorem to expand (2t + 3u)3.
Apply the Binomial Theorem to expand (a + b)3, where a = 2t and b = 3u.
(2t + 3u)3 = 3C0(2t)3(3u)0 + 3C1 (2t)2(3u)1 + 3C2 (2t)1(3u)2 + 3C3(2t)0(3u)3
= 1(8t3)(1) + 3(4t2)(3u) + 3(2t )( 9u2) (1)(27u3)
= 8t3 + 36t2u + 54tu2 + 27u3
Answer: 8t3 + 36t2u + 54tu2 + 27u3
Example 6

36. B. Use the Binomial Theorem to expand (a – 2b2)4.
Expand a Binomial Using the Binomial Theorem
B. Use the Binomial Theorem to expand (a – 2b2)4.
Apply the Binomial Theorem to expand (a + b)4, where a = a and b = –2b2.
(a – 2b2)4 = 4C0 a4(–2b2)0 + 4C1 a3(–2b2) C2 a2(–2b2)2 + 4C3 a1(–2b2) C4 a0(–2b2)4
= 1a4(1) + 4a3(–2b2) + 6a2(4b4) + 4a(–8b6) (1)(16b8)
= a4 – 8a3b2 + 24a2b4 – 32ab6 + 16b8
Answer: a4 – 8a3b2 + 24a2b4 – 32ab6 + 16b8
Example 6

37. Use the Binomial Theorem to expand (3t2 – 2u)4.
A. 81t 8 – 216t 6u + 216t 4u 2 – 96t 2u3 + 16u4
B. 81t 8 – 54t 6u + 36t4u2 – 24t 2u3 + 16u4
C. 81t t 6u + 216t4u2 + 96t 2u3 + 16u 4
D. 81t 4 – 54t 3u + 36t 2u 2 – 24tu u 4
Example 6

38. Represent the expansion of (3x – 5y)17 using sigma notation.
Write a Binomial Expansion Using Sigma Notation
Represent the expansion of (3x – 5y)17 using sigma notation.
Apply the Binomial Theorem to represent the expansion of (a + b)17 using sigma notation, where a = 3x and b = –5y.
Answer:
Example 7

39. Represent the expansion of (2y – 6z)18 using sigma notation.B. C. D.
Example 7

40. LESSON 10–5
The Binomial Theorem