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Example 11. Resolve into factors : 8 x^3  36 x^2 y 54 xy^2  27 y^3.

Solution : 8 x^3  36 x^2 y 54 xy^2  27 y^3

= ( 2 x)^3  3 u( 2 x)^2 u 3 y 3 u 2 xu( 3 y)^2 ( 3 y)^3

= ( 2 x 3 y)^3 ( 2 x 3 y)( 2 x 3 y)( 2 x 3 y)

(g) Applying the formulae : a^3 b^3 (ab)(a^2 abb^2 ) and

a^3 b^3 (ab)(a^2 abb^2 ):

Example 12. Resolve into factors : (i) 8 a^3  27 b^3 (ii)a^6  64

Solution : (i) 8 a^3  27 b^3 ( 2 a)^3 ( 3 b)^3

= ( 2 a 3 b){( 2 a)^2  2 au 3 b( 3 b)^2 }

= ( 2 a 3 b)( 4 a^2  6 ab 9 b^2 )

(ii) a^6 ^64 = (a^2 )^3 ( 4 )^3

= (a^2  4 ){(a^2 )^2 a^2 u 4 ( 4 )^2 }

= (a^2  4 )(a^4  4 a^2  16 )

But, a^2  4 a^2  22 (a 2 )(a 2 )

and a^4  4 a^2  16 (a^2 )^2 ( 4 )^2  4 a^2

= (a^2  4 )^2  2 (a^2 )( 4 ) 4 a^2

= (a^2  4 )^2  4 a^2

= (a^2  4 )^2 ( 2 a)^2

= (a^2  4  2 a)(a^2  4  2 a)

= (a^2  2 a 4 )(a^2  2 a 4 )

Alternative method :

a^6  64 = (a^3 )^2  82

=(a^3  8 )(a^3  8 )

=(a^3  23 )(a^3  23 )

=(a 2 )(a^2  2 a 4 )u(a 2 )(a^2  2 a 4 )

=(a 2 )(a 2 )(a^2  2 a 4 )(a^2  2 a 4 )

( 2 )( 2 )( 2 4 )( 2 4 )

64

2 2

6

     

? 

a a a a a a

a

Activity : Resolve into factors:

1. 2 x^4  16 x 2. 8 a^3  3 a^2 b 3 ab^2 b^3 3. (ab)^3 (ab)^3

(h) Factors of the expression with fractional coefficients :
Factors of the expressions with fraction may be expressed in different ways.

For example, ̧
¹

·

̈

©

§

̧  

¹

·

̈

©

§

  

9

1

3 3

1

3

1

27

(^12)
3
a^3 a^3 a a a
Again, {( 3 ) ( 1 )}
27
1
( 27 1 )
27
1
27
3 1 3 3 3
a  a  a