6 8 The Binomial Theorem Objectives Binomial Expansion

6 -8 The Binomial Theorem

Objectives Binomial Expansion & Pascal’s Triangle The Binomial Theorem

Bellringer In your notebook, solve the following problems 2 1. (a + b) = 3 2. (a + b) = 4 3. (a + b) =

Vocabulary Pascal’s Triangle (a + b) 0 (a + b) 1 (a + b) 2 (a + b) 3 (a + b) 4 (a + b) 5

How to Find Pascal’s Triangle Binomial Expansion Example

Using Pascal’s Triangle Use Pascal’s Triangle to expand (a + b)5. Use the row that has 5 as its second number. The exponents for a begin with 5 and decrease. 1 a 5 b 0 + 5 a 4 b 1 + 10 a 3 b 2 + 10 a 2 b 3 + 5 a 1 b 4 + 1 a 0 b 5 The exponents for b begin with 0 and increase. In its simplest form, the expansion is a 5 + 5 a 4 b + 10 a 3 b 2 + 10 a 2 b 3 + 5 ab 4 + b 5.

Expanding a Binomial Use Pascal’s Triangle to expand (x – 3)4. First write the pattern for raising a binomial to the fourth power. 1 4 6 4 1 Coefficients from Pascal’s Triangle. (a + b)4 = a 4 + 4 a 3 b + 6 a 2 b 2 + 4 ab 3 + b 4 Since (x – 3)4 = (x + (– 3))4, substitute x for a and – 3 for b. (x + (– 3))4 = x 4 + 4 x 3(– 3) + 6 x 2(– 3)2 + 4 x(– 3)3 + (– 3)4 = x 4 – 12 x 3 + 54 x 2 – 108 x + 81 The expansion of (x – 3)4 is x 4 – 12 x 3 + 54 x 2 – 108 x + 81.

Vocabulary You can use the Binomial Theorem to also expand a binomial. I will demonstrate on the board. You can follow along on page 354 of your book.

Using the Binomial Theorem Use the Binomial Theorem to expand (x – y)9. Write the pattern for raising a binomial to the ninth power. (a + b)9 = 9 C 0 a 9 + 9 C 1 a 8 b + 9 C 2 a 7 b 2 + 9 C 3 a 6 b 3 + 9 C 4 a 5 b 4 + 9 C 5 a 4 b 5 + 9 C 6 a 3 b 6 + 9 C 7 a 2 b 7 + 9 C 8 ab 8 + 9 C 9 b 9 Substitute x for a and –y for b. Evaluate each combination. (x – y)9 = 9 C 0 x 9 + 9 C 1 x 8(–y) + 9 C 2 x 7(–y)2 + 9 C 3 x 6(–y)3 + 9 C 4 x 5(–y)4 + 9 C 5 x 4(–y)5 + 9 C 6 x 3(–y)6 + 9 C 7 x 2(–y)7 + 9 C 8 x(–y)8 + 9 C 9(–y)9 = x 9 – 9 x 8 y + 36 x 7 y 2 – 84 x 6 y 3 + 126 x 5 y 4 – 126 x 4 y 5 + 84 x 3 y 6 – 36 x 2 y 7 + 9 xy 8 – y 9 The expansion of (x – y)9 is x 9 – 9 x 8 y + 36 x 7 y 2 – 84 x 6 y 3 + 126 x 5 y 4 – 126 x 4 y 5 + 84 x 3 y 6 – 36 x 2 y 7 + 9 xy 8 – y 9.

Real World Example Dawn Staley makes about 90% of the free throws she attempts. Find the probability that Dawn makes exactly 7 out of 12 consecutive free throws. Since you want 7 successes (and 5 failures), use the term p 7 q 5. This term has the coefficient 12 C 5. Probability (7 out of 10) = 12 C 5 p 7 q 5 = 12! • (0. 9)7(0. 1)5 5! • 7! The probability p of success = 90%, or 0. 9. = 0. 0037881114 Simplify. Dawn Staley has about a 0. 4% chance of making exactly 7 out of 12 consecutive free throws.

Homework Pg 355 # 1, 2, 13, 14