Topic 1.5 - Binomial Expansions

TOPIC 1.5 : BINOMIAL EXPANSION OF (a + b)n, where n < 0 or n = fractions Formula for binomial series:  n  n n (a  b) n  a n   a n1b   a n 2b 2  ...   a n r b r  ...  b n , r  2 1 n n! where n is a positive integer and    n Cr = r !(n  r )! r Example 1: Expand (a) (1  x) 6 (b) 1  2 x (c)

6 in ascending power of x as far as (1  2 x) 3

the term in x3. 1.5.1 Binomial expansion for power n is negative integer or fraction If n is a fraction or negative integer, then 2 n   b n b   bx  n(n  1)  bx   n n (a + bx) = a (1  x) = a (1  x) = a 1  n      ... 2!  a  a a    a   , provided |bx| < |a|.

n

n(n  1) 2 n(n  1)(n  2) 3 n(n  1)(n  2)...( n  r ) r x  x  ... + x +… 2! 3! r! ,where n is a real number, and |x| < 1.

(1 + x)n = 1 + nx +



Example 2: Find the binomial expansion of 4  3 x 2



1

up to and including the term in x4.

1.5.2 Approximations We can use binomial expansion to find numerical approximations to square roots and other calculations, provided x is much smaller than 1. Example 3: 1

Expand (8  3 x) 3 in ascending power of x as far as the term in x3, stating the values of x for which the expansion is valid. Hence obtain an approximate value for

3

8.72 .

1.5.3 Further binomial expansions Example 4

Assuming that x is so small that terms in x3 and higher may be neglected, find a quadratic approximation to 1  x ,stating the values of x for which your answer is valid. 1  2x

Example 5 2 Given the function f(x), 4 x  3 x  2 . ( x  1)( x 2  2) (i) Express f(x) in partial functions. (ii) Hence show that when x is sufficiently small that terms in x4 and higher powers of x to be neglected, 1 7 f ( x )  1  x  x 2  x 3 .

2

4

Ex 5: Binomial Expansions 1 4

1. Expand (1  2 x) in ascending powers of x as far as the term in x3, stating the values of x for 1

which the expansion is valid. Hence obtain approximate value for: (a) 4 1.4 , (b) 1.08 4 . 1

2. Expand 1  3 x  2 in ascending powers of x, up to and including the term in x3, simplifying the coefficients. [Nov 2004 – 4 marks] 

1

3. Expand

in ascending powers of x, up to and including the term in x2, simplifying

2  x  the coefficients. [June 2002 – 4 marks] 3

Expand the given function in ascending powers of x up to the term in x3. 1 1 (b) 1  x (c) 1  2 x (d) (e) (a) 1  2 x 2 2 1  2x 1 x  x 2  x  1  4x 1 x  x2 4.

Assuming that x is so small that terms in x3 and higher may be neglected, find a quadratic approximation to the given function, stating the values of x for which your answer is valid. 1 (a) 1  x (b) 3 8  x (c) 1  3x 1 x (1  x)(3  x) 5.

6.

Given the function f(x),

6  7x . (2  x)(1  x 2 )

(i) Express f(x) in partial functions. (ii) Hence show that when x is sufficiently small for x4 and higher powers of x to be neglected, 1 15 (Nov 2002 P3) 3  5x  x 2  x 3 . 2

4

7.

(Nov 2005 P3) 8.

(Jun 2006 P3) 9.

(Jun 2004 P3) 10.

f(x) ≈