Factors-10

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Factorize :

(i) (1 - x 2 ) (1 - y 2 ) + 4 x y

(ii) x 4 + x 2 y 2 + y 4

(iii) (p 2 + q 2 - r 2 ) 2 - 4 p 2 q 2

Solution :

(i) (1 - x 2 ) (1 - y 2 ) + 4 x y

= 1 - x 2 - y 2 + x 2 y 2 + 4 x y

= 1 + x 2 y 2 + 2 x y - x 2 - y 2 + 2 x y

= (1 + x 2 y 2 + 2 x y) - (x 2 + y 2 - 2 x y)

= (1 + x y) 2 - (x - y) 2

= (1 + x y + x - y) (1 + x y - x + y)

(ii) x 4 + x 2 y 2 + y 4

= x 4 + y 4 + 2 x 2 y 2 - x 2 y 2

= (x 2 + y 2 ) 2 - (x y) 2

= (x 2 + y 2 + x y) (x 2 + y 2 - x y)

(iii) (p 2 + q 2 - r 2 ) 2 - 4 p 2 q 2

= (p 2 + q 2 - r 2 ) 2 - (2 p q) 2

= (p 2 + q 2 - r 2 + 2 p q) (p 2 + q 2 - r 2 - 2 p q)

= ((p 2 + q 2 + 2 p q - r 2 ) ((p 2 + q 2 - 2 p q- r 2 )

= [ (p + q) 2 - r 2 ] [ (p - q) 2 - r 2 ]

= (p + q + r) (p + q - r) (p - q + r) (p - q - r)

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