LESSON 10–5 The Binomial Theorem

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

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

A. 5–Minute Check 1

B. 5–Minute Check 1

C. 5–Minute Check 1

D. 5–Minute Check 1

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

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

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)

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

binomial coefficients Pascal’s triangle Binomial Theorem Vocabulary

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 6 + 1 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

(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. 1 5 10 10 5 1 5th row 1 6 15 20 15 6 1 6th row (a + b)6 = 1a6b0 + 6a5b1 + 15a4b2+ 20a3b3 + 15a2b4 + 6a1b5 + 1a0b6 (a + b)6 = a6 + 6a5b + 15a4b2+ 20a3b3 + 15a2b4 + 6ab5 + b6 Simplify. Example 1

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

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 5 + 1 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

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)2 + 10(3x)2(2)3 + 5(3x)1(2)4 + 1(3x)0(2)5 (3x + 2)5 = 243x5 + 810x4 + 1080x3 + 720x2 + 240x + 32 Answer: 243x5 + 810x4 + 1080x3 + 720x2 + 240x + 32 Example 1

Use Pascal’s Triangle to expand (2x + 3)6. A. 64x6 + 576x5 + 2160x4 + 4320x3 + 162x2 + 486x + 729 B. 64x6 + 96x5 + 144x4 + 216x3 + 324x2 + 486x + 729 C. 64x6 + 576x5 + 2160x4 + 4320x3 + 4860x2 + 2916x + 729 D. 32x5 + 48x4 + 72x3 + 108x2 + 162x + 243 Example 1

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)2 + 20x3(–2y)3 + 15x2(–2y)4 + 6x1(–2y)5 + 1x0(–2y)6 = x6 – 12x5y + 60x4y2 – 160x3y3 + 240x2y4 – 192xy5 + 64y6 Example 2

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

Use Pascal’s Triangle to expand (3x – y)6. A. 729x6 – 243x5y + 81x4y2 – 27x3y3 + 9x2y4 – 3xy5 + y6 B. 729x6 – 1458x5y + 1215x4y2 – 540x3y3 + 135x2y4 – 18xy5 + y6 C. 729x6 + 1458x5y + 1215x4y2 + 540x3y3 + 135x2y4 + 18xy5 + y6 D. 3x6 – 18x5y + 45x4y2 – 60x3y3 + 45x2y4 – 18xy5 + y6 Example 2

Key Concept 3

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

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

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

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

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

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

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

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

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

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

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

Key Concept 6

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(1)(27u3) = 8t3 + 36t2u + 54tu2 + 27u3 Answer: 8t3 + 36t2u + 54tu2 + 27u3 Example 6

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)1 + 4C2 a2(–2b2)2 + 4C3 a1(–2b2)3 + 4C4 a0(–2b2)4 = 1a4(1) + 4a3(–2b2) + 6a2(4b4) + 4a(–8b6) + 1(1)(16b8) = a4 – 8a3b2 + 24a2b4 – 32ab6 + 16b8 Answer: a4 – 8a3b2 + 24a2b4 – 32ab6 + 16b8 Example 6

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 8 + 216t 6u + 216t4u2 + 96t 2u3 + 16u 4 D. 81t 4 – 54t 3u + 36t 2u 2 – 24tu 3 + 16u 4 Example 6

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

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

LESSON 10–5 The Binomial Theorem