Level – 01
1 ) Factorise : x2 + 7x + 12
Solution : x2 + 7x + 12
= x2 + (4 + 3 ) x + 12
= x2 + 4x + 3x + 12
= x ( x + 4 ) + 3 ( x + 4 )
= ( x + 4 ) ( x + 3 )
2 ) Factorise : x2 – x – 12
Solution : x2 – x – 12
= x2 – ( 4 – 3 )x – 12
= x2 – 4x + 3x- 12
= x ( x – 4 ) + 3 ( x – 4 )
= ( x + 4 ) ( x + 3 )
3 ) Factorise : x2 + 7x + 6
Solution : x2 + 7x + 6
= x2 + ( 6 + 1 )x + 6
= x2 + 6x + x + 6
= x ( x + 6 ) + 1 ( x + 6 )
= ( x + 6 ) ( x + 1 )
4 ) Factorise : x2 + 9x + 8
Solution : x2 + 9x + 8
= x2 + ( 8 + 1 )x + 8
= x2 + 8x + x + 8
= x ( x + 8 ) + 1 ( x + 8 )
= ( x + 8 ) ( x + 1 )
5 ) Factorise : x2 – 2x – 8
Solution : x2 – 2x – 8
= x2 – ( 4 – 2 ) x – 8
= x2 – 4x + 2x – 8
= x ( x – 4 ) + 2 ( x – 4 )
= ( x – 4 ) ( x + 2 )
6 ) Factorise : x2 + 7x – 8
Solution : x2 + 7x – 8
= x2 + ( 8 – 1 )x – 8
= x2 + 8x – x – 8
= x ( x – 8 ) – 1 ( x – 8 )
= ( x – 8 ) ( x – 1 )
7 ) Factorise : x2 – 7x – 8
Solution : x2 – 7x – 8
= x2 – ( 8 – 1 ) x – 8
= x2 – 8x + x – 8
= x ( x – 8 ) + 1 ( x – 8 )
= ( x – 8 ) ( x + 1 )
8 ) Factorise : x2 + 40x – 129
Solution : x2 + 40x – 129
= x2 + ( 43 – 3 )x – 129
= x2 + 43x -3x – 129
= x ( x + 43 ) – 3 ( x + 43 )
= ( x + 43 ) ( x – 3 )
9 ) Factorise : x2 + 46x + 129
Solution : x2 + 46x + 129
= x2 + ( 43 + 3 ) x + 129
= x2 + 43x + 3x + 129
= x ( x + 43 ) + 3 ( x + 43 )
= ( x + 43 ) ( x + 3 )
10 ) Factorise : x2 + 7x + 12
Solution : x2 + 7x + 12
= x2 + ( 4 +3 ) x + 12
= x2 +4x + 3x + 12
= x ( x + 4 ) + 3 ( x + 4 )
= ( x + 4 ) ( x + 3 )
11 ) Factorise : x2 + 12x + 35
Solution : x2 + 12x + 35
= x2 + ( 7 + 5 )x + 35
= x2 + 7x + 5x + 35
= x ( x + 7 ) + 5 ( x + 7 )
= ( x + 7 ) ( x + 5 )
12 ) Factorise : x2 + 2x – 35
Solution : x2 + 2x – 35
= x2 + ( 7 – 5 )x – 35
= x2 + 7x – 5x – 35
= x ( x + 7 ) – 5 ( x + 7 )
= ( x + 7 ) ( x – 5 )
13 ) Factorise : x2 – 2x – 35
Solution : x2 – 2x – 35
= x2 – ( 7 – 5 )x – 35
= x2 – 7x + 5x – 35
= x ( x – 7 ) + 5 ( x – 7 )
= ( x- 7 ) ( x + 5 )
14 ) Factorise : x2 + 36x + 35
Solution : x2 + 36x + 35
= x2 + ( 35 + 1 )x + 35
= x2 + 35x + x + 35
= x ( x + 35 ) + 1 ( x + 35 )
= ( x + 35 ) ( x + 1 )
15 ) Factorise : x2 – 36x + 35
Solution : x2 – 36x + 35
= x2 – ( 35 + 1 )x + 35
= x2 – 35x – x + 35
= x ( x – 35 ) – 1 ( x – 35 )
= ( x – 35 ) ( x – 1 )
Level – 02
1 ) ( a + b )2 – 5a – 5b + 6
= ( a + b )2 – 5(a + b) + 6
Let ( a + b ) = m
Then, the given algebraic expression becomes,
m2 -5m + 6
= m2 – ( 3 + 2 )m + 6
= m2 – 3m – 2m + 6
= m ( m – 3 ) – 2( m – 3 )
= ( m – 3 ) ( m – 2 )
Now, putting m = ( a + b ), we get
= ( a + b -3 ) ( a + b – 2 )
2 ) ( x2 – 2x )2 + 5 ( x2 – 2x )2 – 36
Let ( x2 – 2x )2 = p, then the given algebraic expression becomes,
p2 + 5p – 36
= p2 + ( 9 – 4 )p – 36
=p2 + 9p – 4p – 36
= p( p + 9 ) – 4( p + 9 )
= ( p + 9 ) ( p – 4 )
Now, putting p = ( x2 – 2x )2 , we get
= ( x2 – 2x + 9 ) ( x2 – 2x – 4 )
3 ) ( p2 – 3q2 ) 2 – 16 ( p2 – 3q2 ) + 63
Let p2 – 3q2 = a, then the given algebraic expression becomes
a2 – 16 a + 63
= a2 – ( 7+ 9 ) a + 63
= a2 – 7a – 9a + 63
= a ( a – 7 ) – 9 ( a – 7 )
= ( a – 7 ) ( a – 9 )
Putting a = p2 – 3q2 , we get
= ( p2 – 3q2 – 7 ) ( p2 – 3q2 – 9 )
Level – 03
4 ) a4 + 4a2 – 5
= a4 + ( 5 – 1 )a2 – 5
= a4 + 5a2 – a2 – 5
= a2 ( a2 + 5 ) – 1( a2 + 5 )
= ( a2 + 5 )( a2 – 1 )
= ( a2 + 5 ){(a)2 – ( 1 )2}
= ( a2 + 5 ) ( a + 1 ) ( a – 1 )
5 ) a6 + 3a3b3 – 40b6
= a6 + ( 8 – 5 ) a3b3 – 40b6
= a6 + 8a3b3 – 5a3b3 – 40b6
= a3 ( a3 + 8b3 ) – 5b3 ( a3 – 8b3 )
= ( a3 + 8 ) ( a3 – 5b3 )
= {(a)3 + (2 )3 } ( a3 -5b3 )
= ( a + 2 ){ (a)2 -(a) (2) + (2)2} ( a3 – 5b3 )
= ( a + 2 )( a2 – 2a + 4 )( a3 – 5b3 )
Level – 04
6 ) ( x + 1 )( x + 3 )( x – 4 )( x – 6 ) + 24
= { ( x + 1 )( x – 4 )}{( x + 3 )(x – 6 )} + 24
= ( x2 – 4x + x – 4 )( x2 + 3x – 6x – 18 ) + 24
= ( x2 – 3x – 4 ) ( x2 – 3x – 18 ) + 24
Let x2 – 3x = a, then
( a – 4 ) ( a – 18 ) + 24
= a2 – 4a – 18a + 72 + 24
= a2 – 22 a + 96
= a2 – ( 16 + 6 ) a + 96
= a 2 – 16a – 6a + 96
= a( a – 16 ) – 6 ( a – 16)
= ( a – 16 ) ( a – 6 )
Putting a = x2 – 3x, we get
= ( x2 – 3x – 16 ) ( x2 – 3x – 6 )
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