= ( 3 1 )( 9 3 1 )
27
(^12)
a a a
Here, in the second solution, the factors involving the variables are with integral
coefficients. This result can be expressed as the first solution :
( 3 1 )( 9 3 1 )
27
(^12)
a a a
= ( 9 3 1 )
9
1
( 3 1 )
3
(^12)
a u a a
= ̧
¹
·
̈
©
§
̧
¹
·
̈
©
§
9
1
3 3
1 2 a
a a
Example 13. Resolve into factors : x^3 6 x^2 y 11 xy^2 6 y^3.
Solution : x^3 6 x^2 y 11 xy^2 6 y^3
= {x^3 3 x^2 2 y 3 x( 2 y)^2 ( 2 y)^3 }xy^2 2 y^3
= (x 2 y)^3 y^2 (x 2 y)
= (x 2 y){(x 2 y)^2 y^2 }
= (x 2 y)(x 2 yy)(x 2 yy)
= (x 2 y)(x 3 y)(xy)
= (xy)(x 2 y)(x 3 y)
Activity : Resolve into factors:
3
1
6
7
2
(^12)
x x 2.
8
3 1
a 3. 16 x^2 25 y^2 8 xz 10 yz
Exercise 3⋅ 3
Resolve into factors (1 to 43) :
a^2 abacbc 2. abab 1
(xy)(xy)(xy)(yz)(xy)(zx) 4. ab(xy)bc(xy)
9 x^2 24 x 16 6. a^4 27 a^2 1
x^4 6 x^2 y^2 y^4 8. (a^2 b^2 )(x^2 y^2 ) 4 abxy
4 a^2 12 ab 9 b^2 4 c^2 10. 9 x^4 45 a^2 x^2 36 a^4
a^2 6 a 8 y^2 2 y 12. 16 x^2 25 y^2 8 xz 10 yz
2 b^2 c^2 2 c^2 a^2 2 a^2 b^2 a^4 b^4 c^4 14. x^2 13 x 36
x^4 x^2 20 16. a^2 30 a 216