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Binomial Theorem: Simplifying Powers and Expressions

Dive into the Binomial Theorem in math class! Learn to expand powers of binomials easily with patterns and Pascal's Triangle. Explore the coefficients and combinatorial forms using examples like (a + b)2 and (a + b)3. Practice factoring and homework to deepen your understanding.

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Binomial Theorem: Simplifying Powers and Expressions

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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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