ENGINEERING MATHEMATICS 4 BINOMIAL EXPANSION BA501 1 CHAP 1 - Binomial Expansions (Kembangan Binomial) The binomial theorem describes the algebraic expansion of powers of a binomial. Figure 1 : Example use the binomial Expansion in geometric ENGINEERING MATHEMATICS 4 BINOMIAL EXPANSION BA501 2 There are 3 methods to expand binomial expression Method 1 - Algebra method Expansion two or more expression. Example: The expansion depend on power value (n) n = 0, (a + x)0 = 1 n = 1, (a + x)1 = a + x n = 2, (a + x)2 = (a + x) (a + x) = a2 + 2ax + x2 n = 3, (a + x)3 = (a + x) (a + x) (a + x) = a3 + 3a2x + 3ax2 + x3 n = 4, (a + x)4 = (a + x)(a + x)(a + x)(a + x) = a4 +4a3x +6a2x2 +4ax3+ x4 Method 2 - PASCAL Triangle Pascal's triangle is a triangular array of the binomial coefficients in a triangle. It is named after the French mathematician Blaise Pascal Base on algebra method. only using the coefficients of terms. Power value Coefficient n = 0 1 n = 1 1 1 n = 2 1 2 1 n = 3 1 3 3 1 n = 4 1 4 6 4 1 n = 5 1 5 10 10 5 1 n = 6 1 ? 1 ENGINEERING MATHEMATICS 4 BINOMIAL EXPANSION BA501 3 Example: (1 + 2x)5 n = 5 1 5 10 10 5 1 (1 + 2x)5 = 5 0 4 3 2 2 3 4 0 5) 2 ( ) 1 ( 1 ) 2 )( 1 ( 5 ) 2 ( ) 1 ( 10 ) 2 ( ) 1 ( 10 ) 2 ( ) 1 ( 5 ) 2 ( ) 1 ( 1 x x x x x x + + + + + = x x x x x 32 80 80 40 10 1 + + + + + Method 3 - Binomial theorem Sum of terms (Hasil tambah sebutan) The general terms = nCr an - rxr With r = 0,1,2,3,4 How to use calculator to calculate nCr nCr Example: Find value of 6C3 using calculator 6 C 3 = 20 nCr = ( ) ! !!r r nn n value SHIFT r value n value SHIFT r value = ENGINEERING MATHEMATICS 4 BINOMIAL EXPANSION BA501 4 The expansion of (a + x)n where n is a positive integer. The expansion of (a + x)n where n is a positive integer, power for first term a is n and power for increase terms are (n-1) example : n = 5 (a + x)n = an + 5a(n-1)x + 10a(n-2)x2 + 10a(n-3)x3 + 5a(n-4) x4 + x5 (a + x)5 = a5 + 5a4x + 10a3x2 + 10a2x3 + 5ax4 + x5 The general terms of Binomial Expansions: Formula 1 : (a + x)n = an + nC1an - 1x + nC2an - 2x2 + nC3an - 3x3 + ..+ nCrxn - rar + ... + xn = an + nan - 1x + n(n - 1)an - 2x2 + n(n - 1)(n - 2)an- 3x3 + ... + 1x2 1x2x3 n(n - 1)(n - 2) (n - r + 1)(an - rxr) + ... + xn 1x2x3x.... r ENGINEERING MATHEMATICS 4 BINOMIAL EXPANSION BA501 5 Example 1: Expand the expression using binomial theorem a) (1 + 3x)4 b) ( 3 2x )3 ENGINEERING MATHEMATICS 4 BINOMIAL EXPANSION BA501 6 c) (2 +4x )5 d) ( x 2y)6 ENGINEERING MATHEMATICS 4 BINOMIAL EXPANSION BA501 7 e) ( 2a b)4 ENGINEERING MATHEMATICS 4 BINOMIAL EXPANSION BA501 8 Coefficients value and terms value for positive integer n Example 2 : 1. Find the coefficient of x3 and 5th term of the binomial expansion of ( )65 3 x . a = , x = , n = General term = nCr an - rxr where r = 0,1,2,3,4,. r = term -1 ENGINEERING MATHEMATICS 4 BINOMIAL EXPANSION BA501 9 2. Find the term include x3 and the term independent of x in the expansion 1221|.|

\| xx . a = , x = , n = Exercise : 1. Expand the expression: a) 421 |.|

\| +x 2. Find the 3rd term in the expansion below: a) 921|.|

\| xx b) 6235 ||.|

\|x 3. Find the coefficient value for x18 of expansion ( )1233x x + ENGINEERING MATHEMATICS 4 BINOMIAL EXPANSION BA501 10 Binomial expansion of (1+x)n where n (power) is not a positive integer ( not positive integer means fraction or negative value ) In this case the Binomial expansions an infinite series. Condition: - Expansion in form ( 1+x )n - This series is infinite - If expression in form (a + x)n the value of a 1, hence expansion must use formula 2 General term If value of a in binomial expression not 1 (a 1) Use formula 2 nx) 1 ( += + + ++ + ......4 3 2 1) 3 )( 2 )( 1 (3 2 1) 2 )( 1 (2 1) 1 (14 3 2xx x xn n n nxx xn n nxxn nnx Expansion a 1 (a + x)n = nnnaxaaxa |.|

\| + =((

|.|

\| + 1 1 nnaxa |.|

\| + 1 = (((

+ |.|

\| + |.|

\| + |.|

\| + + ...4 3 2 1) 3 )( 2 )( 1 (3 2 1) 2 )( 1 (2 1) 1 (14 3 2axx x xn n n naxx xn n naxxn naxn ENGINEERING MATHEMATICS 4 BINOMIAL EXPANSION BA501 11 Example : In ascending power of x, obtain the first 4 terms of : (a) 1) 2 ( x (b) x + 41 ENGINEERING MATHEMATICS 4 BINOMIAL EXPANSION BA501 12 (c) ( )32 82x ENGINEERING MATHEMATICS 4 BINOMIAL EXPANSION BA501 13 d) 22231||.|

\|x Exercise : Find the expansion of expression below up to the term in x3 : a) ( )311 x + b) ( )122 4 + x c) ( )2127 9 x + ENGINEERING MATHEMATICS 4 BINOMIAL EXPANSION BA501 14 Approximations Use Binomial Theorem If 3) 05 . 2 ( = + = (2 + 0.05)3 Example : 1. Obtain the approximation value to 4 decimal places a) (1.08)5 = (1 + 0.08)5 , ( a = x = n = ) use (a + x)n = an + nC1an - 1x + nC2an - 2x2 + nC3an - 3x3 + ... + nCrxn - rar + ... + xn (1 + 0.08)5 = Whole number x = 2.05 whole number ENGINEERING MATHEMATICS 4 BINOMIAL EXPANSION BA501 15 b) ( )2997 . 01 ENGINEERING MATHEMATICS 4 BINOMIAL EXPANSION BA501 16 2. Find the binomial expansion of ( )52 1 x +. Hence find the value of ( )50024 . 1 to 5 places of decimals ENGINEERING MATHEMATICS 4 BINOMIAL EXPANSION BA501 17 3. Using Binomial Expansion, find first 4 terms of ( )102 x +. Hence find the approximation value of ( )1098 . 1until 4 decimal places. Expansion Method for approximation Note : If value of x is so small or mark slightly we can use formula ( 1+x )n 1 + nx exp : (1+ 0.00002)6 = 1 + 6(0.00002) = 1 + 0.00012 = 1.00012 (2+ 0.00006)3 = (2 + 0.00006)3 = 23(1 + 3(0.00003)) = 8.00072 ENGINEERING MATHEMATICS 4 BINOMIAL EXPANSION BA501 18 Approximation for multiplication and division example : 1. Find the value of expression below to 5 decimal places using binomial theorem: i) (1.01)3(96 . 0) ii) ( )22102 . 0 106 . 1 ENGINEERING MATHEMATICS 4 BINOMIAL EXPANSION BA501 19 iii) ( ) ( )( )231402 . 0 199 . 0 07 . 1 iv) Find the value of (5.04)4 to four places of decimals using binomial theorem. ENGINEERING MATHEMATICS 4 BINOMIAL EXPANSION BA501 20 v) Using binomial theorem, find the value of 72526|.|

\| to four decimal places. ENGINEERING MATHEMATICS 4 BINOMIAL EXPANSION BA501 21 Exercise: 1. Find the binomial expansion of each of the following : a) ( )43 2 y x + b) 524 |.|

\| +x b) ( )42 1 x d) 432 |.|

\| x 2. Find the term include x4 and the term independent of x in the expansion 1231|.|

\| xx. 3. For the Binomial Expansion 166221|.|

\| +xx,find coefficient value of x8 . 4. Find the expansion of ( )( )313 112xx+ 5. Expand ( )311 x + up to terms x3. Hence using the expansion before, find the approximation value of 3(1.08) to 5 places of decimals. 6. Find the approximation value for 932 . 01 7. Using binomial theorem, find the value of ( ) ( )3388 . 092 . 0 01 . 1 to four decimal places.