Master the Binomial Expansion Theorem in MATH 106
Understand the Binomial Expansion Theorem with examples, coefficient calculations, and historical notes in MATH 106, Section 9. Get ready for the quiz with provided homework hints.
Master the Binomial Expansion Theorem in MATH 106
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Section 9Binomial Expansion • Questions about homework? • Submit homework! Recall the exercises we did last class: MATH 106, Section 9
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
#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
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
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
#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
#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
#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
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
