Cambridge Additional Mathematics

¡n

2

¢

=n(n¡1)

2

for all integersn> 2.

(2)

(1)

276 Counting and the binomial expansion (Chapter 10)

Example 20 Self Tutor

Find the coefficient of x^5 in the expansion of (x+ 3)(2x¡1)^6.

(x+ 3)(2x¡1)^6

=(x+ 3)[(2x)^6 +

¡ 6

1

¢

(2x)^5 (¡1) +

¡ 6

2

¢

(2x)^4 (¡1)^2 +::::]

=(x+ 3)(2^6 x^6 ¡

¡ 6

1

¢

25 x^5 +

¡ 6

2

¢

24 x^4 ¡::::)

So, the terms containingx^5 are

¡ 6

2

¢

24 x^5 from (1)

and ¡ 3

¡ 6

1

¢

25 x^5 from (2)

) the coefficient ofx^5 is

¡ 6

2

¢

24 ¡ 3

¡ 6

1

¢

25 =¡ 336

12 Find the coefficient ofx^5 in the expansion of (x+ 2)(x^2 +1)^8.

13 Find the term containingx^6 in the expansion of (2¡x)(3x+1)^9.

14 Find the coefficient ofx^4 in the expansion of:

a (3¡ 2 x)^7 b (1 + 3x)(3¡ 2 x)^7

15 Find:

a the coefficient ofx^7 in the expansion of (x^2 ¡3)(2x¡5)^8

b the term independent ofxin the expansion of (1¡x^2 )

³

x+

2

x

́ 6

.

16 When the expansion of (a+bx)(1¡x)^6 is written in ascending powers ofx, the first three terms are

3 ¡ 20 x+cx^2. Find the values ofa,b, andc.

Example 21 Self Tutor

Consider the expansion of (1 + 3x)n, where n 2 Z+.

If the coefficient ofx^2 is 90 , find the value ofn.

(1 + 3x)n has general term Tr+1=

¡n

r

¢

1 n¡r(3x)r

=

¡n

r

¢

3 rxr

) T 3 =

¡n

2

¢

32 x^2 is thex^2 term.

Since the coefficient ofx^2 is 90 ,

¡n

2

¢

£9=90

)

n(n¡1)

2

=10

) n^2 ¡n=20

) n^2 ¡n¡20 = 0

) (n¡5)(n+4)=0

) n=5 fn> 0 g

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Additional Mathematics
Y:\HAESE\CAM4037\CamAdd_10\276CamAdd_10.cdr Monday, 6 January 2014 12:03:39 PM BRIAN