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Agenda – Thurs. May 24 (Day 2)

Agenda – Thurs. May 24 (Day 2). The Binomial Theorem (Sec. 5.4) Homework Unit 4-5 Test: Friday! Objective: “To learn how to expand powers of binomials – the easy way!”. 5.4 The Binomial Theorem. Binomial – two terms. Expand the following: (a + b) 2 (a + b) 3 (a + b) 4.

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Agenda – Thurs. May 24 (Day 2)

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  1. Agenda – Thurs. May 24(Day 2) • The Binomial Theorem (Sec. 5.4) • Homework • Unit 4-5 Test: Friday! • Objective: “To learn how to expand powers of binomials – the easy way!”

  2. 5.4 The Binomial Theorem

  3. Binomial – two terms • Expand the following: • (a + b)2 • (a + b)3 • (a + b)4 Study each answer. Is there a pattern that we can use to simplify our expressions?

  4. Notice that each entry in the triangle corresponds to a value nCr : 0C0 1C0 1C1 2C02C12C2 3Co3C13C23C3 n! (n-r)!r! So, we can see that tn,r = nCr =

  5. By Pascal’s triangle we can see that… nCr = n-1Cr-1 + n-1Cr Rewrite the following using Pascal’s Triangle: a) 10C4 b)18C8 + 18C9 =19C9 =10-1C4-1 + 10-1C4 Reverse the process for one of the terms =9C3 + 9C4

  6. The coefficients of each term in the expansion of (a + b)n correspond to the terms in row “n” of Pascal’s Triangle. Therefore you can write these coefficients in combinatorial form. Lets look at (2a + 3b)3 = 8a3 + 36a2b +54ab2 + 27b3 Notice that there is one more term than the exponent number!

  7. (a + b)n = nC0an + nC1an-1b+ nC2an-2b2 + … + nCran-rbr + … + nCnbn or Expand: (a + b)5 Try it with (3x – 2y)4

  8. Factoring using the binomial theorem: Rewrite 1 + 10x2 + 40x4 + 80x6 + 80 x8 + 32x10 in the form (a + b)n Step 1: We know that there are 6 terms so the exponent must be 5. Step 2: The first term is an, which is found by taking the 5th root of 1 Therefore, a = 1 Step 3: The final term is 32x10 …So you have to rewrite this term with a power of 5. Therefore, b = (2x2)5 = 2x2 Therefore, the factored form is (1 + 2x2)5

  9. Homework Read: pg. 289 - 293 Do: pg. 293 # 1ace, 3ab, 4ab (you may need to draw the triangle first), 5bcd, 8, 9ace,11ad,12a, 16a, 21

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