Expand each of the following, using suitable identities:
(i) 
(ii) 
(iii) 
(iv) 
(v) 
(vi)
(i) Using identity,
(a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca
Here, a = x, b = 2y and c = 4z
(x + 2y + 4z)2 = x2 + (2y)2 + (4z)2 + (2 * x * 2y) + (2 * 2y * 4z) + (2 * 4z * x)
= x2 + 4y2 + 16z2 + 4xy + 16yz + 8xz
(ii) Using identity,
(a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca
Here, a = 2x, b = -y and c = z
(2x – y + z)2 = (2x)2 + (-y)2 + z2 + (2 * 2x * -y) + (2 * y * z) + (2 * z * 2x)
= 4x2 + y2 + z2 – 4xy – 2yz + 4xz
(iii) Using identity, (a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca
Here, a = -2x, b = 3y and c = 2z
(-2x + 3y + 2z)2 = (-2x)2 + (3y)2 + (2z)2 + (2 * 2x * 3y) + (2 * 3y * 2z) + (2 * 2z * 2x)
= 4x2 + 9yz + 4z2 – 12xy + 12yz – 8xz
(iv) Using identity, (a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca
Here, a = 3a, b = -7b and c = -c
(3a – 7b – c)2 = (3a)2 + (-7b)2 + (-c)2+ (2 *3a * -7b) + (2 * -7b * -c) + (2 * -c * 3a)
= 9a2 + 49b2 + c2 – 42ab + 14bc – 6ac
(v) Using identity, (a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca
Here, a = -2x, b = -5y and c = -3z
(-2x + 5y – 3z)2 = (-2x)2 + (-5y)2 + (-3z)2+ (2 *-2x * 5y) + (2 * 5y * -3z) + (2 * -3z * -2x)
= 4x2 + 25y2 + 9z2 – 20xy -30yz + 12xz
(vi) Using identity, (a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca
Here, a = 
a, b = 
b and c = 1
(
a – 
b + 1)2 = (
a)2 + (-
b)2 + (1)2+ (2 *
a * -
b) + (2 * -
b * 1) + (2 * 1 * 
a)
= 
a2 + 
b2 + 1 – 
ab - b + 
a