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


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