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NAME 10-5 DATE PERIOD Study Guide and Intervention The Binomial Theorem Pascal’s Triangle In Pascal’s triangle, the first and last numbers in each row is 1 and the number in row 0 is 1. Other numbers are the sum of the two numbers above them. The first five rows of Pascal’s triangle are shown below. Row 0 1 Row 1 1 Row 2 1 Row 3 Row 4 1 1 1 2 3 1 3 4 6 1 4 1 The numbers in Pascal’s triangles are the binomial coefficients when (a + b)n is expanded. You can use these numbers to expand binomials without multiplying repeatedly. The first term is an, the last term is bn, and the powers of a decrease by 1 as the powers of b increase by 1 from left to right. Example Use Pascal’s triangle to expand (x + 2y) 5. First, write the series for (a + b)5 without coefficients. Then replace a with x and b with 2y. a5b0 + a4b1 + a3b2 + a2b3 + a1b4 + a0b5 Series for (a + b)5 5 0 4 1 3 2 2 3 1 4 0 5 x (2y) + x (2y) + x (2y) + x (2y) + x (2y) + x (2y) Substitution 5 4 3 2 2 3 4 5 x + x (2y) + x (4y ) + x (8y ) + x (16y ) + 32y Simplify. Exercises Use Pascal’s triangle to expand each binomial. 1. (x + 4)3 2. (3x + y)4 x3 + 12x2 + 48x + 64 3. (7 + g)4 81x4 + 108x3y + 54x2y2 + 12xy3 + y4 2401 + 1372g + 294g2 + 28g3 + g4 4. (m - n)6 m6 - 6m5n + 15m4n2 - 20m3n3 + 15m2n4 - 6mn5 + n6 5. (2a – 2b)5 32a5 - 160a4b + 320a3b2 - 320a2b3 + 160ab4 - 32b5 6. (c + d)7 Chapter 10 c7 + 7c6d + 21c5d2 + 35c4d3 + 35c3d4 + 21c2d5 + 7cd6 + d7 26 Glencoe Precalculus Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. The numbers in the fifth row of Pascal’s triangle are the coefficients. Following the pattern above, these numbers will be 1, 5, 10, 10, 5, and 1. 1x5 + 5x4 (2y) + 10x3 (4y2) + 10x2 (8y3) + 5x (16y4) + 1 · 32y5 Add coefficients. 5 4 3 2 2 3 4 5 x + 10x y + 40x y + 80x y + 80xy + 32y Simplify. NAME DATE 10-5 Study Guide and Intervention PERIOD (continued) The Binomial Theorem The Binomial Theorem The binomial coefficient of the an - r br term in the expansion of (a + b) is given by nCr. You can find nCr by using a n n! calculator or by finding − . (n - r)! r! The Binomial Theorem states that for any positive integer n, the expansion of (a + b)n is C anb0 + nC1 an - 1b1 + nC2 an - 2b2 + … + nCr an - rbr + … + nCn a0bn. n 0 Example 1 Find the coefficient of the fourth term in the expansion of (5a + 2b)6. For (5a + 2b)6 to have the form (a + b)n, let a = 5a and b = 2b. Since r increases from 0 to n, r is one less than the number of the term. Evaluate 6C3. 6! 6! · 5 · 4 · 3! C3 = − =− = 6− or 20 6 (6 - 3)!3! 3!3! 3!3! Example 2 Use the Binomial Theorem to expand (3x + 7)4. Let a = 3x and b = 7. (3x + 7)4 = 4C0(3x)4(7)0 + 4C1(3x)3(7)1 + 4C2(3x)2(7)2 + 4C3(3x)1(7)3 + 4C4(3x)0(7)4 = 1 · 81x4 · 1 + 4 · 27x3 · 7 + 6 · 9x2 · 49 + 4 · 3x · 343 + 1 · 1 · 2401 = 81x4 + 756x3 + 2646x2 + 4116x + 2401 Exercises Find the coefficient of the indicated term in each expansion. 1. (x + 5)6, fourth term 2. (3a + 4b)8, a3b5 term 2500 1,548,288 Use the Binomial Theorem to expand each binomial. 3. (x + 3)5 4. (4x + 2y)3 x5 + 15x4 + 90x3 + 270x2 + 405x + 243 5. (x - 2y)4 64x3 + 96x2y + 48xy2 + 8y3 6. (2x - 3y)4 x4 - 8x3y + 24x2y2 - 32xy3 + 16y4 Chapter 10 16x4 - 96x3y + 216x2y2 - 216xy3 + 81y4 27 Glencoe Precalculus Lesson 10-5 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. The binomial coefficient of the fourth term in (a + b)6 is 20. Substitute for a and b in an - rbr. 20(5a)6 – 3(2b)3 = 20(5a)3(2b)3 = 20(125a3)(8b) = 20,000a3b The coefficient is 20,000.