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Transcript
Name 5-7 Class Date Reteaching The Binomial Theorem You can find the coefficients of a binomial expansion in Pascal’s Triangle. To create Pascal’s Triangle, start by writing a triangle of 1’s. This triangle forms the first two rows. Each row has one more element than the one above it. Each row begins with a 1, and then each element is the sum of the two closest elements in the row above. The last element in each row is a 1. Pascal’s Triangle 1 2 1 3 6 4 1 1 3 1 1 Problem 1 1 5 1 4 10 1 10 5 1 What is the expansion of (x 1 y)5 ? Use Pascal’s Triangle. Step 1 The power of the binomial corresponds to the second number in each row of Pascal’s Triangle. Because the power of this binomial is 5, use the row of Pascal’s Triangle with 5 as the second number. The numbers of this row are the coefficients of the expansion. Pascal’s Triangle 1 1 1 1 1 1 1 2 3 4 5 1 3 6 10 1 1 4 10 5 1 Step 2 The exponents of the x-terms of the expansion Count down for x. begin with the power of the binomial and decrease to 0. The exponents of 1x 5y 0! 5x 4y1 ! 10x 3y2 ! 10x 2y3 ! 5x 1y 4 ! 1x 0y5 the y-terms of the expansion begin with 0 and increase to the power of Count up for y. the binomial. Step 3 Simplify all terms to write the expansion in standard form. (x 1 y)5 5 x5 1 5x4y 1 10x3y2 1 10x2y3 1 5xy4 1 y5 Exercises Write the expansion of each binomial. 1. (a 1 b)3 a3 1 3a2b 1 3ab2 1 b3 2. (x 2 y)4 x4 2 4x3y 1 6x2y2 2 4xy3 1 y4 3. (r 1 1)5 r5 1 5r4 1 10r3 1 10r2 1 5r 1 1 4. (a 2 b)6 a6 2 6a5b 1 15a4b2 2 20a3b3 1 15a2b4 2 6ab5 1 b6 Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 69 Name Class 5-7 Date Reteaching (continued) The Binomial Theorem • The Binomial Theorem states that for any binomial (a 1 b) and any positive integer n, (a 1 b)n 5 nC0an 1 nC1an21b 1 nC2an22b2 1 . . . 1 nCn21abn21 1 nCnbn. • The theorem provides an effective method for expanding any power of a binomial. n! . Evaluate the combination nCk as k!(n 2 k)! Problem What is the expansion of (3x 1 2)3 ? Use the Binomial Theorem. Step 1 Determine a, b, and n. a 5 3x, b 5 2, n 5 3 Step 2 Use the formula to write the equation. (3x 1 2)3 5 3C0(3x)3 1 3C1(3x)2(2) 1 3C2(3x)(2)2 1 3C3(2)3 Step 3 Simplify. 5 1(27x3) 1 3(9x2)(2) 1 3(3x)(4) 2 1(8) 5 27x3 1 54x2 1 36x 1 8 Exercises Fill in the correct coefficients, variables, and exponents for the expanded form of each binomial. u 1 ux3y 1 6xuy2 1 uxyu 1 u4 4; 4; 2; 4; 3; y 6. (z 2 y)3 5 zu 2 u z2y 1 u zyu 2 u3 3; 3; 3; 2; y 7. (x 1 z)5 5 xu 1 ux4z 1 10xuz2 1 ux2zu 1 uxz4 1 u5 5; 5; 3; 10; 3; 5; z 5. (x 1 y)4 5 x Write the expansion of each binomial. Use the Binomial Theorem. 8. (x 1 y)5 9. (x 2 y)5 x5 1 5x4y 1 10x3y2 1 10x2y3 1 5xy4 1 y5 10. (2x 1 y) 3 8x3 1 12x2y 1 6xy2 1 y3 12. (x 2 2y)5 14. (x 2 3y)4 11. (x 1 3y)4 x4 1 12x3y 1 54x2y2 1 108xy3 1 81y4 13. (2x 2 y) 5 32x5 2 80x4y 1 80x3y2 2 40x2y3 1 10xy4 2 y5 15. (4x 2 y)3 x5 2 10x4y 1 40x3y2 2 80x2y3 1 80xy4 2 32y5 x4 2 12x3y 1 54x2y2 2 108xy3 1 81y4 16. (x 2 x5 2 5x4y 1 10x3y2 2 10x2y3 1 5xy4 2 y5 64x3 2 48x2y 1 12xy2 2 y3 1)5 17. (1 2 x)3 x5 2 5x4 1 10x3 2 10x2 1 5x 2 1 18. (x2 1 1) 3 x6 1 3x4 1 3x2 1 1 1 2 3x 1 3x2 2 x3 19. (y2 1 a)4 y8 1 4y6a 1 6y4a2 1 4y2a3 1 a4 Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 70