Question

Factorise 

1.  \(4p^ 2 - 9q ^2\) 
2.  \(63a^ 2 - 112b ^2\) 
3.  \(49x^ 2 - 36\) 
4.  \(16x^ 5 - 144x^ 3\)
5.  \((l + m)^ 2 - (l - m) ^2\) 
6.  \(9x^ 2 y^ 2 - 16 \)
7.  \((x^ 2 - 2xy + y^ 2 ) - z^ 2\) 
8.  \(25a ^2 - 4b^ 2 + 28bc - 49c ^2\)

Answer 1

(i) \(4p^ 2 - 9q^ 2 = (2p) ^2 - (3q)^ 2\) 

= \((2p + 3q) (2p - 3q) [a ^2 - b ^2\) = \((a - b) (a + b)]\)

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(ii) \(63a^ 2 - 112b ^2\) 

= \(7(9a^ 2 - 16b^ 2 )\) 

= \(7[(3a)^2 - (4b) ^2 ]\) 

= \(7(3a + 4b) (3a - 4b) [a^ 2 - b^ 2= (a - b) (a + b)]\)

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(iii) \(49x ^2 - 36\) 

= \((7x)^ 2 - (6)^2\) 

= \((7x - 6) (7x + 6) [a^ 2 - b^ 2 = (a - b) (a + b)]\)

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(iv) \(16x ^5 - 144x ^3\) 

= \(16x^ 3 (x^ 2 - 9)\) 

= \(16 x^ 3 [(x)^ 2 - (3)^2 ]\) 

= \(16 x^ 3 (x - 3) (x + 3) [a^ 2 - b ^2 = (a - b) (a + b)]\)

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(v) \((l + m) ^2 - (l - m) ^2\) 

= \([(l + m) - (l - m)] [(l + m) + (l - m)] [Using \;identity \;a ^2 - b ^2 = (a - b) (a + b)]\) 

= \((l + m - l + m) (l + m + l - m)\) 

= \(2m x ^2l = 4ml = 4lm\)

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(vi) \(9x^ 2 y^ 2 - 16\) 

= \((3xy) ^2 - (4)^2\) 

= \((3xy - 4) (3xy + 4) [a ^2 - b ^2 = (a - b) (a + b)]\)

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(vii) \((x^ 2 - 2xy + y ^2 ) - z^ 2\) 

= \((x - y) ^2 - (z) ^2 [(a - b)^ 2 = a ^2 - 2ab + b ^2 ]\) 

= \((x - y - z) (x - y + z) [a^ 2 - b^ 2 = (a - b) (a + b)]\)

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(viii) \(25a ^2 - 4b^ 2 + 28bc - 49c^ 2\) 

= \(25a^ 2 - (4b^ 2 - 28bc + 49c ^2 )\) 

= \((5a)^ 2 - [(2b)^ 2 - 2 \times 2b \times 7c + (7c)^2 ]\) 

= \((5a) ^2 - [(2b - 7c)^2 ] [Using \;identity (a - b) ^2 = a ^2 - 2ab + b ^2 ]\) 

= \([5a + (2b - 7c)] [5a - (2b - 7c)] [Using \;identity \;a^ 2 - b ^2 = (a - b) (a + b)]\) 

= \((5a + 2b - 7c) (5a - 2b + 7c)\)