Factorise the following expressions.
(i) a 2 + 8a + 16
(ii) p 2 − 10p + 25
(iii) 25m 2 + 30m + 9
(iv) 49y 2 + 84yz + 36z 2
(v) 4x 2 − 8x + 4
(vi) 121b 2 − 88bc + 16c 2
(vii) (l + m)2 − 4lm (Hint: Expand (l + m)2 first)
(viii) a 4 + 2a 2 b 2 + b 4
(i) a2 + 8a + 16 = (a)2 + 2 × a × 4 + (4)2
= (a + 4)2 [(x + y)2 = x2 + 2xy + y2]
(ii) p2 − 10p + 25 = (p)2 − 2 × p × 5 + (5)2
= (p − 5)2 [(a − b)2 = a2 − 2ab + b2]
(iii) 25m2 + 30m + 9 = (5m)2 + 2 × 5m × 3 + (3)2
= (5m + 3)2 [(a + b)2 = a2 + 2ab + b2]
(iv) 49y2 + 84yz + 36z2 = (7y)2 + 2 × (7y) × (6z) + (6z)2
= (7y + 6z)2 [(a + b)2 = a2 + 2ab + b2]
(v) 4x2 − 8x + 4 = (2x)2 − 2 (2x) (2) + (2)2
= (2x − 2)2 [(a − b)2 = a2 − 2ab + b2]
= [(2) (x − 1)]2 = 4(x − 1)2
(vi) 121b2 − 88bc + 16c2 = (11b)2 − 2 (11b) (4c) + (4c)2
= (11b − 4c)2 [(a − b)2 = a2 − 2ab + b2]
(vii) (l + m)2 − 4lm = l2 + 2lm + m2 − 4lm
= l2 − 2lm + m2
= (l − m)2 [(a − b)2 = a2 − 2ab + b2]
(viii) a4 + 2a2b2 + b4 = (a2)2 + 2 (a2) (b2) + (b2)2
= (a2 + b2)2 [(a + b)2 = a2 + 2ab + b2]