6-8 The Binomial Theorem.

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Presentation transcript:

6-8 The Binomial Theorem

Objectives Binomial Expansion & Pascal’s Triangle The Binomial Theorem

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

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

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. 1a5b0 + 5a4b1 + 10a3b2 + 10a2b3 + 5a1b4 + 1a0b5 The exponents for b begin with 0 and increase. In its simplest form, the expansion is a5 + 5a4b + 10a3b2 + 10a2b3 + 5ab4 + b5.

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 = a4 + 4a3b + 6a2b2 + 4ab3 + b4 Since (x – 3)4 = (x + (–3))4, substitute x for a and –3 for b. (x + (–3))4 = x4 + 4x3(–3) + 6x2(–3)2 + 4x(–3)3 + (–3)4 = x4 – 12x3 + 54x2 – 108x + 81 The expansion of (x – 3)4 is x4 – 12x3 + 54x2 – 108x + 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 = 9C0a9 + 9C1a8b + 9C2a7b2 + 9C3a6b3 + 9C4a5b4 + 9C5a4b5 + 9C6a3b6 + 9C7a2b7 + 9C8ab8 + 9C9b9 Substitute x for a and –y for b. Evaluate each combination. (x – y)9 = 9C0x9 + 9C1x8(–y) + 9C2x7(–y)2 + 9C3x6(–y)3 + 9C4x5(–y)4 + 9C5x4(–y)5 + 9C6x3(–y)6 + 9C7x2(–y)7 + 9C8x(–y)8 + 9C9(–y)9 = x9 – 9x8y + 36x7y2 – 84x6y3 + 126x5y4 – 126x4y5 + 84x3y6 – 36x2y7 + 9xy8 – y9 The expansion of (x – y)9 is x9 – 9x8y + 36x7y2 – 84x6y3 + 126x5y4 – 126x4y5 + 84x3y6 – 36x2y7 + 9xy8 – y9.

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 p7q5. This term has the coefficient 12C5. Probability (7 out of 10) = 12C5 p7q5 = • (0.9)7(0.1)5 The probability p of success = 90%, or 0.9. 12! 5! •7! = 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