Factorise the following expressions.

(i)


(ii)


(iii)


(iv)


(v)


(vi)


(vii) (Hint: Expand first)


(viii)

(i) a2 + 8a + 16


= a2 + 2× a × 4 + 42


Using identity (a + b)2 = a2 + 2ab + b2


Here a =a; b = 4


a2 + 2× a × 4 + 42


= (a + 4)2


= (a + 4)(a + 4)


 


(ii) p2 -10p + 25


= p2 - 2× 5 × p + 52


Using identity (a - b)2 = a2 - 2ab + b2


Here a =p; b = 5


p2 - 2× 5 × p + 52


= (p - 5)2


= (p-5)(p-5)


 


(iii) 25m2 + 30m + 9


= (5m)2 + 2 × 5 × 3 × m + 32


Using identity (a + b)2 = a2 + 2ab + b2


Here a =5m; b = 3


(5m)2 + 2 × 5 × 3 × m + 32


= (5m + 5)2


= (5m + 5)(5m + 5)


 


(iv) 49y2 + 84yz + 36z2


= (7y)2 + 2 × 7 × 6 × y × z + (6z)2


Using identity (a + b)2 = a2 + 2ab + b2


Here a =7y; b = 6z


(7y)2 + 2 × 7 × 6 × y × z + (6z)2


= (7y + 6z)2


(7y + 6z)(7y + 6z)


(v) 4x2 - 8x + 4


= (2x)2 - 2 × 2 × 2× x + 22


Using identity (a - b)2 = a2 - 2ab + b2


Here a =2x; b = 2


(2x)2 - 2 × 2 × 2× x + 2


= (2x - 2)2


= (2x - 2)(2x - 2)


 


(vi) 121b2 - 88bc + 16c2


= (11b)2 - 2 × 11b × 4c + (4c)2


Using identity (a - b)2 = a2 - 2ab + b2


Here a =11b; b = 4c


(11b)2 - 2 × 11b × 4c + (4c)2


= (11b – 4c)2


(11b – 4c)(11b – 4c)


(vii) (l + m)- 4lm 


Expand (l + m)2 = l2 + 2lm + m2


[using (a + b)2 = a2 + 2ab + b2]


l2 + 2lm + m2 - 4lm


l2 - 2lm + m2


= l2 - 2 × l × m + m2


Using identity (a - b)2 = a2 - 2ab + b2


Here a =l; b = m


l2 - 2 × l × m + m2


= (l – m)2


= ( l - m)(l - m)


 


(viii) a⁴ + 2a2b2 + b⁴


= (a2)2 + 2 × a2 × b2 + (b2)2


Using identity (a + b)2 = a2 + 2ab + b2


Here a = a2; b = b2


(a2)2 + 2 × a2 × b2 + (b2)2


= (a2 + b2)2


(a2 + b2)(a2 + b2)

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