Cambridge Additional Mathematics

272 Counting and the binomial expansion (Chapter 10)

Example 16 Self Tutor

Find the:

a 5 th row of Pascal’s triangle b binomial expansion of

³

x¡

2

x

́ 5

.

a 1

11

121

1331

14641

1 5 10 10 5 1

the 0 th row, for (a+b)^0

the 1 st row, for (a+b)^1

the 5 th row, for (a+b)^5

b Using the coefficients obtained ina, (a+b)^5 =a^5 +5a^4 b+10a^3 b^2 +10a^2 b^3 +5ab^4 +b^5

Letting a=(x) and b=

³

¡ 2

x

́

, we find

³

x¡

2

x

́ 5

=(x)^5 +5(x)^4

³

¡ 2

x

́

+ 10(x)^3

³

¡ 2

x

́ 2

+ 10(x)^2

³

¡ 2

x

́ 3

+5(x)

³

¡ 2

x

́ 4

+

³

¡ 2

x

́ 5

=x^5 ¡ 10 x^3 +40x¡

80

x

+

80

x^3

¡

32

x^5

EXERCISE 10F

1 Use the binomial expansion of (a+b)^3 to expand and simplify:

a (p+q)^3 b (x+1)^3 c (x¡3)^3

d (2 +x)^3 e (3x¡1)^3 f (2x+5)^3

g (2a¡b)^3 h

¡

3 x¡^13

¢ 3

i

³

2 x+

1

x

́ 3

2 Use (a+b)^4 =a^4 +4a^3 b+6a^2 b^2 +4ab^3 +b^4 to expand and simplify:

a (1 +x)^4 b (p¡q)^4 c (x¡2)^4

d (3¡x)^4 e (1 + 2x)^4 f (2x¡3)^4

g (2x+b)^4 h

³

x+

1

x

́ 4

i

³

2 x¡

1

x

́ 4

3 Expand and simplify:

a (x+2)^5 b (x¡ 2 y)^5 c (1 + 2x)^5 d

³

x¡

1

x

́ 5

4 Expand and simplify (2 +x)^5 +(2¡x)^5.

5aWrite down the 6 th row of Pascal’s triangle.

b Find the binomial expansion of:

i (x+2)^6 ii (2x¡1)^6 iii

³

x+

1

x

́ 6

6 Expand and simplify:

a (1 +

p

2)^3 b (

p

5+2)^4 c (2¡

p

2)^5

cyan magenta yellow black

(^05255075950525507595)
100 100
(^05255075950525507595)
100 100 4037 Cambridge
Additional Mathematics
Y:\HAESE\CAM4037\CamAdd_10\272CamAdd_10.cdr Monday, 23 December 2013 4:33:01 PM BRIAN