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Section 9 Binomial Expansion

Section 9 Binomial Expansion. Questions about homework? Submit homework!. Recall the exercises we did last class:. What is “Binomial Expansion”?. #1. Take an expression of the form ( x + y ) n and multiply it out for n = 2 and for n = 3. ( x + y ) 2 =

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Section 9 Binomial Expansion

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  1. Section 9Binomial Expansion • Questions about homework? • Submit homework! Recall the exercises we did last class: MATH 106, Section 9

  2. What is “Binomial Expansion”? #1 Take an expression of the form (x + y)n and multiply it out for n = 2 and for n = 3. (x + y)2 = Before simplification, how many terms did we get? (x + y)3 = Before simplification, how many terms did we get? yx + x2 + 2xy + y2 x2 + y2 = (x + y)(x + y) = xy + 4 = 22 (x + y)(x + y)(x + y) = yyx + xyy + yxy + x3 + xyx + yxx + xxy + y3 = x3 + 3x2y + 3xy2 + y3 8 = 23 RECIPE FOR CHOOSING ONE TERM IN THE EXPANSION OF (x + y)n BEFORE SIMPLIFICATION: Choose x or y from first factor and then choose x or y from second factor and then … MATH 106, Section 9

  3. #2 Without doing any multiplication, find the following expansion: (x + y)7 = If we did the multiplication, how many terms would we get before simplification? x7 + x6y + x5y2 + x4y3 +x3y4 +x2y5 +xy6 +y7 21 7 7 35 35 21 128 = 27 RECIPE FOR CHOOSING ONE TERM IN THE EXPANSION OF (x + y)n BEFORE SIMPLIFICATION: Choose x or y from first factor and then choose x or y from second factor and then … Observe that the coefficients in the simplified expansion of (x + y)n match the row of Pascal’s Triangle corresponding to n. MATH 106, Section 9

  4. The Binomial Theorem (bottom of page 67 in the textbook) (x + y)n = C(n, 0)xny0 + C(n, 1)xn-1y1 + C(n, 2)xn-2y2 + C(n, 3)xn-3y3 + … + C(n, n-2)x2yn-2 + C(n, n-1)x1yn-1 + C(n, n)x0yn Because of symmetry, we could choose to write the formula in the Binomial Theorem as follows: (x + y)n=C(n, n)xny0 +C(n, n-1)xn-1y1 +C(n, n-2)xn-2y2 +C(n, n-3)xn-3y3 + … + C(n, 2)x2yn-2 + C(n, 1)x1yn-1 + C(n, 0)x0yn MATH 106, Section 9

  5. HISTORICAL NOTE: Sometimes the notation C(n, k) is written as Both forms are sometimes called the “binomial coefficient.” Let’s do some! MATH 106, Section 9

  6. #3 • Find the expansion of each of the following: • (x + y)4 • (x + 3)4 • (x2 + 3)4 • (x – y)4 x4 + x3y + x2y2 + xy3 +y4 6 4 4 x4 + x3(3) + x2(3)2 + x(3)3 + (3)4 = 6 4 4 x4 + 12x3 + 54x2 + 108x + 81 (x2)4 + (x2)3(3) + (x2)2(3)2 + (x2)(3)3 + (3)4 = 6 4 4 x8 + 12x6 + 54x4 + 108x2 + 81 x4 + x3(–y) + x2(–y)2 + x(–y)3 + (–y)4 = 6 4 4 x4– 4x3y + 6x2y2– 4xy3 +y4 MATH 106, Section 9

  7. #4 • Find the expansion of each of the following: • (x + 3y)4 • (x2 – 2)6 • (3x4 + 2y3)5 • (3x4 – 2y3)5 x4 + x3(3y) + x2(3y)2 + x(3y)3 +(3y)4 = 6 4 4 x4 + 12x3y + 54x2y2 + 108xy3 + 81y4 (x2)6 + (x2)5(2) + (x2)4(2)2 + (x2)3(2)3 + (x2)2(2)4 + (x2)(2)5 + (2)6 = 20 15 6 15 6 x12 12x10 + 60x8 160x6 + 240x4 192x2 + 64 10 5 10 (3x4)5 + (3x4)4(2y3) + (3x4)3(2y3)2 + (3x4)2(2y3)3 + (3x4)(2y3)4 + (2y3)5 = 5 243x20 + 810x16y3 + 1080x12y6 + 720x8y9 + 240x8y9 + 32y15 243x20 810x16y3 + 1080x12y6 720x8y9 + 240x8y9 32y15 MATH 106, Section 9

  8. #5 • Determine the coefficient of x6y4in the expansion of (x + y)10 • Determine the coefficient of x12y4in the expansion of (x2 + 2y)10 • In the expansion of (x4– 3y3)9 , determine the coefficient of • x20y12 • x24y9 210 x6y4 C(10, 4) x6y4 = 210 (x2)6(2y)4 = C(10, 4) (x2)6(2y)4 = 210 x1224y4 = 3360 x12y4 5 4 10206 x20y12 126 x20(–3)4y12 = –3y3 C( ) ( ) ( ) = x4 9, 4 6 3 –2268 x24y9 –3y3 84 x24(–3)3y9 = x4 9, 3 C( ) ( ) ( ) = MATH 106, Section 9

  9. Homework Hints: In Section 9 Homework Problems #3, 4, 5, and 6, In Section 9 Homework Problems #7, 8, and 9, do not actually do the algebraic expansions. Instead, use the Binomial Theorem and Pascal’s Triangle. Also, don’t forget that ( y)n is equal to yn if n is even, and ( y)n is equal to yn if n is odd. be sure to use Problem #4 on the Section #9 Class Handout as a guide. Quiz #2 NEXT CLASS! Be sure to do the review problems for this, quiz posted on the internet. The link can be found in the course schedule. MATH 106, Section 9

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