# 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 + 9y2 + 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

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