Expand each of the following, using suitable identities:
Expand each of the following, using suitable identities:
(i) (x + 2y + 4z)2 (ii) (2x – y + z)2 (iii) (–2x + 3y + 2z)2
(iv) (3a – 7b – c)2 (v) (–2x + 5y – 3z)2 (vi) [1/4 a - 1/2 b + 1]2
Answer
(i) (x + 2y + 4z)2
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) (2x – y + z)2
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) (–2x + 3y + 2z)2
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 + 9y2 + 4z2 - 12xy + 12yz - 8xz
(iv) (3a – 7b – c)2
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) (–2x + 5y – 3z)2
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) [1/4 a - 1/2 b + 1]2
Using identity, (a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca
Here, a = 1/4 a, b = -1/2 b and c = 1
[1/4 a - 1/2 b + 1]2 = (1/4 a)2 + (-1/2 b)2 + 12 + (2×1/4 a×-1/2 b) + (2×-1/2 b×1) + (2×1×1/4 a)
= 1/16 a2 + 1/4 b2 + 1 - 1/4 ab - b + 1/2 a
