* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download 1. Expand (a b)n Using Pascal`s Triangle Section 15.4 The Binomial
Survey
Document related concepts
List of important publications in mathematics wikipedia , lookup
History of trigonometry wikipedia , lookup
Vincent's theorem wikipedia , lookup
Wiles's proof of Fermat's Last Theorem wikipedia , lookup
Nyquist–Shannon sampling theorem wikipedia , lookup
Fermat's Last Theorem wikipedia , lookup
Brouwer fixed-point theorem wikipedia , lookup
Central limit theorem wikipedia , lookup
Pythagorean theorem wikipedia , lookup
Factorization wikipedia , lookup
Karhunen–Loève theorem wikipedia , lookup
Fundamental theorem of calculus wikipedia , lookup
Fundamental theorem of algebra wikipedia , lookup
Binomial coefficient wikipedia , lookup
Transcript
mes96990_ch15_918-926.qxd 11/9/10 2:52 PM Page 918 BIA— 918 Chapter 15 Sequences and Series Section 15.4 The Binomial Theorem Objectives 1. 2. 3. 4. 5. Expand (a ⴙ b)n Using Pascal’s Triangle Evaluate Factorials n Evaluate a b r Expand (a ⴙ b)n Using the Binomial Theorem Find a Specified Term in the Expansion of (a ⴙ b)n In this section, we will learn how to expand a binomial, (a b) n, where n is a nonnegative integer. We first encountered expansion of binomials in Chapter 6 when we learned to expand binomials such as (a b) 2 and (a b) 3. To expand (a b) 2 means to find the product (a b)(a b). Multiplying the binomials using FOIL gives us (a b) 2 (a b)(a b) a2 ab ab b2 a2 2ab b2 Multiply using FOIL. To expand (a b) 3 means to find the product (a b)(a b)(a b) or (a b) (a b) 2. Expanding (a b) 3 gives us (a b) 3 (a b)(a b)(a b) (a b)(a b) 2 (a b)(a2 2ab b2 ) a3 2a2b ab2 a2b 2ab2 b3 a3 3a2b 3ab2 b3 Distribute. Combine like terms. Expanding, for example, (a b) 5 in this way would be an extremely long process. There are other ways to expand binomials, and the first one we will discuss is Pascal’s triangle. 1. Expand (a ⴙ b)n Using Pascal’s Triangle Here are the expansions of (a b) n for several values of n: (a b) 0 1 (a b) 1 a b (a b) 2 a2 2ab b2 (a b) 3 a3 3a2b 3ab2 b3 (a b) 4 a4 4a3b 6a2b2 4ab3 b4 (a b) 5 a5 5a4b 10a3b2 10a2b3 5ab4 b5 Notice the following patterns in the expansion of (a b) n: 1) There are n 1 terms in the expansion of (a b) n. For example, in the expansion of (a b) 4, n 4 and the expansion contains 4 1 5 terms. 2) The first term is an and the last term is bn. 3) Reading the expansion from left to right, the exponents on a decrease by 1 from one term to the next, while the exponents on b increase by 1 from one term to the next. 4) In each term in the expansion, the sum of the exponents of the variables is n. The coefficients of the terms in the expansion follow a pattern too. If we write the coefficients in triangular form, we obtain Pascal’s triangle, named after seventeenth-century French mathematician Blaise Pascal. The numbers in the nth row of the triangle tell us the coefficients of the terms in the expansion of (a b) n. Coefficients of the Terms in the Expansion of: (a b) : (a b) 1: (a b) 2: (a b) 3: (a b) 4: (a b) 5: Pascal’s Triangle 0 1 1 1 1 1 2 3 1 3 1 1 4 6 4 1 1 5 10 10 5 1 etc. mes96990_ch15_918-926.qxd 11/9/10 2:52 PM Page 919 BIA— Section 15.4 The Binomial Theorem 919 Notice that the first and last numbers of each row in the triangle are 1. The other numbers in the triangle are obtained by adding the two numbers above it. For example, here is how to obtain the sixth row from the fifth: Fifth row (n 4): 1 4 6 4 1 Sixth row (n 5): 1 5 10 10 5 1 Example 1 Expand (a b) 6. Solution The coefficients of the terms in (a b) 5 are given by the last row of the triangle above. We must find the next row of the triangle to find the coefficients of the terms in the expansion of (a b) 6. (a b) 5: 1 5 10 10 5 1 (a b) 6: 1 6 15 20 15 6 1 Recall that the first term is an and the last term is bn. Since n 6, the first term will be a6, and the exponent of a will decrease by 1 for each term. The variable b will appear in the second term and increase by 1 for each term until the last term, b6. (a b) 6 a6 6a5b 15a4b2 20a3b3 15a2b4 6ab5 b6 ■ You Try 1 Expand (a b) 7. Although Pascal’s triangle is a better way to expand (a b) n than doing repeated polynomial multiplication, it can be tedious for large values of n. A more practical way to expand a binomial is by using the binomial theorem. Before learning this method, we need to learn about factorials and binomial coefficients. 2. Evaluate Factorials The notation n! is read as “n factorial.” Definition n! n(n 1)(n 2)(n 3) . . . (1) , where n is a positive integer. Note By definition, 0! 1. mes96990_ch15_918-926.qxd 11/9/10 2:52 PM Page 920 BIA— 920 Chapter 15 Sequences and Series Example 2 Evaluate. a) 4! b) 7! Solution a) 4! 4 ⴢ 3 ⴢ 2 ⴢ 1 24 b) 7! 7 ⴢ 6 ⴢ 5 ⴢ 4 ⴢ 3 ⴢ 2 ⴢ 1 5040 ■ You Try 2 Evaluate. a) 3! b) 6! n 3. Evaluate a b r Factorials are used to evaluate binomial coefficients. A binomial coefficient has the form n a b, read as “the number of combinations of n items taken r at a time” or as “n choose r.” r n a b is used extensively in many areas of mathematics including probability. Another r n notation for a b is nCr. r Definition Binomial Coefficient n n! a b r r !(n r)! where n and r are positive integers and r n. Example 3 Evaluate. 5 a) a b 3 9 b) a b 2 3 c) a b 3 4 d) a b 0 Solution 5 a) To evaluate a b, substitute 5 for n and 3 for r. 3 n n! a b r r!(n r)! 5 5! a b 3 3!(5 3)! 5! 3!2! 5ⴢ4ⴢ3ⴢ2ⴢ1 (3 ⴢ 2 ⴢ 1)(2 ⴢ 1) Let n 5 and r 3. Subtract. Rewrite each factorial as a product. mes96990_ch15_918-926.qxd 11/9/10 2:52 PM Page 921 BIA— Section 15.4 The Binomial Theorem 921 At this point, do not find the products in the numerator and denominator. Instead, divide out common factors in the numerator and denominator. 5ⴢ4ⴢ3ⴢ2ⴢ1 (3 ⴢ 2 ⴢ 1)(2 ⴢ 1) 20 2 10 Divide out common factors. Multiply. Simplify. 5 a b 10 3 9 b) To evaluate a b, substitute 9 for n and 2 for r. 2 9 9! a b Let n 9 and r 2. 2 2!(9 2)! 9! Subtract. 2! 7! 9ⴢ8ⴢ7ⴢ6ⴢ5ⴢ4ⴢ3ⴢ2ⴢ1 Rewrite each factorial as a product. (2 ⴢ 1)(7 ⴢ 6 ⴢ 5 ⴢ 4 ⴢ 3 ⴢ 2 ⴢ 1) 9ⴢ8ⴢ7ⴢ6ⴢ5ⴢ4ⴢ3ⴢ2ⴢ1 Divide out common factors. (2 ⴢ 1)(7 ⴢ 6 ⴢ 5 ⴢ 4 ⴢ 3 ⴢ 2 ⴢ 1) 72 2 36 9 a b 36 2 3 c) To evaluate a b, substitute 3 for n and for r. 3 n n! a b r!(n r)! r 3 3! a b Let n 3 and r 3. 3 3!(3 3)! 3! Subtract. 3! 0! 3! Divide out common factors; 0! 1. 3!(1) 1 Simplify. 1 1 3 a b1 3 4 d) To evaluate a b, substitute 4 for n and 0 for r. 0 4 4! Let n 4 and r 0. a b 0 0!(4 0)! 4! Subtract. 0! 4! 4! Divide out common factors; 0! 1. (1)(4!) 1 1 Simplify. 1 4 a b1 0 ■ mes96990_ch15_918-926.qxd 11/9/10 2:52 PM Page 922 BIA— 922 Chapter 15 Sequences and Series Note We can extend the results of c) and d) and say that for any natural number n, n n a b 1 and a b 1 n 0 You Try 3 Evaluate. a) 4 a b 1 b) 8 a b 5 c) 3 a b 0 6 d) a b 6 4. Expand (a ⴙ b)n Using the Binomial Theorem Now that we can evaluate a binomial coefficient, we state the binomial theorem for expanding (a b) n. Definition Binomial Theorem: For any positive integer n, n n n n b abn1 bn (a b) n an a b an1b a b an2b2 a b an3b3 . . . a 1 2 3 n1 The same patterns that emerged in the expansion of (a b) n using Pascal’s triangle appear when using the binomial theorem. Keep in mind that 1) there are n 1 terms in the expansion. 2) the first term in the expansion is an and the last term is bn. 3) after an, the exponents on a decrease by 1 from one term to the next, while b is introduced in the second term and then the exponents on b increase by 1 from one term to the next. 4) in each term in the expansion, the sum of the exponents of the variables is n. Example 4 Use the binomial theorem to expand (a b) 4. Solution Let n 4 in the binomial theorem. 4 4 4 (a b) 4 a4 a b a41b a b a42b2 a b a43b3 b4 1 2 3 4 4 4 a4 a b a3b a b a2b2 a b ab3 b4 1 2 3 Notice that the exponents of a decrease by 1 while the exponents of b increase by 1. 4! 3 4! 2 2 4! ab ab ab3 b4 1! 3! 2! 2! 3! 1! a4 4a3b 6a2b2 4ab3 b4 a4 This is the same result as the expansion on p. 918. ■ mes96990_ch15_918-926.qxd 11/9/10 2:52 PM Page 923 BIA— Section 15.4 The Binomial Theorem 923 You Try 4 Use the binomial theorem to expand (a b) 3. Example 5 Use the binomial theorem to expand (x 6) 3. Solution Substitute x for a, 6 for b, and 3 for n in the binomial theorem to expand (x 6) 3. 3 3 (x 6) 3 (x) 3 a b(x) 31 (6) a b(x) 32 (6) 2 (6) 3 1 2 x3 3(x2 )(6) (3)x(36) 216 x3 18x2 108x 216 ■ You Try 5 Use the binomial theorem to expand (y 5) 4. When expanding a binomial containing the difference of two terms, rewrite the expression in terms of addition. Example 6 Use the binomial theorem to expand (2x 3y) 5. Solution Since the binomial theorem applies to the expansion of (a b) n, rewrite (2x 3y) 5 as [2x (3y)] 5. Substitute 2x for a, 3y for b, and 5 for n in the binomial theorem. Be sure to put 2x and 3y in parentheses to find the expansion correctly. 5 5 [2x (3y)] 5 (2x) 5 a b(2x) 51 (3y) a b12x) 52 (3y) 2 1 2 5 5 53 3 a b(2x) (3y) a b(2x) 54 (3y) 4 (3y) 5 3 4 5 4 32x (5)(2x) (3y) (10)(2x) 3 (9y2 ) (10)(2x) 2 (27y3 ) (5)(2x) 1 (81y4 ) (243y5 ) 32x5 (5)(16x4 )(3y) (10)(8x3 )(9y2 ) (10)(4x2 )(27y3 ) (5)(2x)(81y4 ) (243y5 ) 5 32x 240x4y 720x3y2 1080x2y3 810xy4 243y5 ■ You Try 6 Use the binomial theorem to expand (3x 4y) 4. mes96990_ch15_918-926.qxd 11/9/10 2:52 PM Page 924 BIA— 924 Chapter 15 Sequences and Series 5. Find a Specified Term in the Expansion of (a ⴙ b)n If we want to find a specific term of a binomial expansion without writing out the entire expansion, we can use the following formula. Definition The kth Term of a Binomial Expansion: The kth term of the expansion of (a b) n is given by n! ank1bk1 (n k 1)!(k 1)! where k n 1. Example 7 Find the fifth term in the expansion of (c2 2d) 8. Solution Since we want to find the fifth term, k 5, use the formula above with a c2, b 2d, n 8, and k 5. The fifth term is 8! 8! (c2 ) 851 (2d) 51 (c2 ) 4 (2d) 4 (8 5 1)!(5 1)! 4! 4! 70c8 (16d 4 ) 1120c8d 4 You Try 7 Find the sixth term in the expansion of (2m n2 ) 9. Using Technology We will discuss how to compute factorials and the binomial coefficient on a graphing calculator. Sometimes it is quicker to calculate them by hand, and sometimes a calculator will make our work easier. Evaluating 3! can be done very easily by multiplying: 3! 3 2 1 6. To find 10! by hand we would multiply: 10! 10 9 8 7 6 5 4 3 2 1 3,628,800. On a graphing calculator, we could find 10! either by performing this multiplication or we can use a special function. Graphing calculators have a factorial key built in. It is found using the MATH key. When you press MATH , move the arrow over to the PRB column, and you will see this menu: Notice that choice 4 is the factorial symbol. To compute 10!, enter 10 and then press MATH . Highlight PRB so that you see the screen at above right. Choose 4: ! and press ENTER .The screen displays 10!. Press ENTER to see that 10! 3,628,800. ■ mes96990_ch15_918-926.qxd 11/9/10 2:52 PM Page 925 BIA— Section 15.4 The Binomial Theorem 925 n Because the binomial coefficient, a b, is used so often in mathematical applications, most graphing r calculators have a built-in key that performs the calculations for you. It is located on the same menu as the factorial. Refer to the first calculator screen to see that nCr is choice number 3. 9 To find the value of a b, press 9 MATH , and then highlight PRB. 4 Choose 3: nCr and press ENTER . Now enter 4 and press ENTER . The screen will look like the next screen here.The value 9 of a b is 126. 4 Although the calculator has functions to evaluate factorials and binomial coefficients, sometimes it is actually quicker to evaluate them by hand. Think about this as you evaluate the following problems. Evaluate each of the following using the methods discussed in this section. Verify the result using a graphing calculator. Think about which method you prefer for each problem. 1) 4! 2) 6! 5) 7 a b 4 6) a 3) 15 b 8 5 4) a b 2 9! 7) a 18 b 14 8) a 25 b 24 Answers to You Try Exercises 1) a7 7a6b 21a5b2 35a4b3 35a3b4 21a2b5 7ab6 b7 2) a) 6 5) y4 20y3 150y2 500y 625 7) b) 720 3) a) 4 b) 56 c) 1 d) 1 4) a3 3a2b 3ab2 b3 81x4 432x3y 864x2y2 768xy3 256y4 6) 4 10 2016m n Answers to Technology Exercises 1) 24 2) 720 3) 362,880 4) 10 5) 35 6) 6435 7) 3060 8) 25 15.4 Exercises Objective 1: Expand (a ⴙ b)n Using Pascal’s Triangle Objective 2: Evaluate Factorials 9) In your own words, explain how to evaluate n! for any positive integer. 1) In your own words, explain how to construct Pascal’s triangle. 10) Evaluate 0!. 2) What are the first and last terms in the expansion of (a b) n? Evaluate. Use Pascal’s triangle to expand each binomial. 3) (r s)3 4) (m n)4 5) (y z)5 6) (c d)6 7) (x 5)4 8) (k 2)5 VIDEO 11) 2! 12) 3! 13) 5! 14) 6! mes96990_ch15_918-926.qxd 11/9/10 2:52 PM Page 926 BIA— 926 Chapter 15 Sequences and Series Objective 3: Evaluate a b n r Evaluate each binomial coefficient. 5 15) a b 2 4 16) a b 2 7 VIDEO 17) a b 3 8 18) a b 5 19) a 9 20) a b 3 10 b 4 44) (4c 3d) 4 45) (x2 1) 3 46) (w3 2) 3 4 1 47) a m 3nb 2 5 1 48) a a 2bb 3 3 1 49) a y 2z2 b 3 50) at 2 Find the indicated term of each binomial expansion. 9 21) a b 7 22) a 4 23) a b 4 5 24) a b 5 52) ( y 4) 7; fifth term 6 25) a b 1 3 26) a b 1 54) (z 3) 9; seventh term 5 27) a b 0 7 28) a b 0 11 b 8 51) (k 5) 8; third term 53) (w 1) 15; tenth term VIDEO 55) (q 3) 9; second term 56) (u 2) 7; fourth term 57) (3x 2) 6; fifth term 58) (2w 1) 9; seventh term 29) How many terms are in the expansion of (a b) ? 9 30) Before expanding (t 4) using the binomial theorem, how should the binomial be rewritten? 6 59) (2y2 z) 10; eighth term 60) (p 3q2 ) 8; fifth term 61) (c3 3d2 ) 7; third term 62) (2r3 s4 ) 6; sixth term Use the binomial theorem to expand each expression. 63) (5u v3 ) 11; last term 31) ( f g) 64) (4h k4 ) 12; last term 32) (c d) 3 33) (w 2) 5 34) (h 4) 4 4 35) (b 3) 5 36) (t 9) 3 37) (a 3) 4 38) ( p 2) 3 39) (u v) 40) ( p q) 3 41) (3m 2) 4 1 4 ub 2 Objective 5: Find a Specified Term in the Expansion of (a ⴙ b) n Objective 4: Expand (a ⴙ b) n Using the Binomial Theorem VIDEO 43) (3a 2b) 5 5 42) (2k 1) 4 n 65) Show that a b 1 for any positive integer n. n n 66) Show that a b n for any positive integer n. 1