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= x^2 (xy)xy(xy) 2 y^2 (xy)

= (xy)(x^2 xy 2 y^2 )

= (xy)(x^2  2 xyxy 2 y^2 )

= (xy){x(x 2 y)y(x 2 y)}

= (xy)(x 2 y)(xy)

= (xy)^2 (x 2 y)

?(xy) is a factor of g(x)

?x^2 xy 2 y^2

=x^2 xy 2 xy 2 y^2

=x(xy) 2 y(xy)

=(xy)(x 2 y)

?x^3  3 xy^2  2 y^3 (xy)^2 (x 2 y)

Example 3. Resolve into factors : 54 x^4  27 x^3 a 16 x 8 a.

Solution : Let, f(x) 54 x^4  27 x^3 a 16 x 8 a

then, f a a a a a 8 a
2

1

16

2

1

27

2

1

54

2

1

4 3

̧

¹

·

̈

©

 §

̧

¹

·

̈

©

 §

̧

¹

·

̈

©

§

̧

¹

·

̈

©

§

= 8 8 0
8

27

8

(^2744)
a  a  a a
?
2 2
1 a
x a ̧ x
¹
·
̈
©
§ i.e., 2 xa is a factor of f(x).
Now, 54 x^4  27 x^3 a 16 x 8 a 27 x^3 ( 2 xa) 8 ( 2 xa) ( 2 xa)( 27 x^3  8 )
( 2 xa){( 3 x)^3 ( 2 )^3 }=( 2 xa)( 3 x 2 )( 9 x^2  6 x 4 )
Activity : Resolve into factors :
1.x^3  21 x 20 2. 2 x^3  3 x^2  3 x 1 3. x^3  6 x^2  11 x 6
Exercise 3⋅ 4
Resolve into factors :

1. 6 x^2  7 x 1 2. 3 a^3  2 a 5


2. x^3  7 xy^2  6 y^3 4. x^2  5 x 6


3. 2 x^2 x 3 6. 3 x^2  7 x 6


4. x^3  2 x^2  5 x 6 8. x^3  4 x^2 x 6


5. a^3  3 a 36 10. a^4  4 a 3


6. a^3 a^2  10 a 8 12. x^3  3 x^2  4 x 4


7. a^3  7 a^2 b 7 ab^2 b^3 14. x^3 x 24


8. x^3  6 x^2 y 11 xy^2  6 y^3 16. 2 x^4  3 x^3  3 x 2


9. 4 x^4  12 x^3  7 x^2  3 x 2 18. x^6 x^5 x^4 x^3 x^2 x


10. 4 x^3  5 x^2  5 x 1 20. 18 x^3  15 x^2 x 2