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4.Factorize the following using appropriate 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)\([1/4a - 1/2b + 1]^2\)
Answer :

i)\((x + 2y + 4z)^2 = (x)^2 + (2y)^2 +(4z)^2 + 2(x)(2y) + 2(2y)(4z) + 2(x)(4z)\)
by using identity (v)
i.e., \((a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ac\)
\(= x^2 + 4y^2 + 16z^2 + 4xy + 16yz + 8xz\)

ii)\((2x - y + z)^2 = (2x)^2 + (-y)^2 + (z)^2 - 2(2x)(y) - 2(y)(z) + 2(2x)(z)\)
by using identity (v)
i.e., \((a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ac\)
\(= 4x^2 + y^2 + z^2 - 4xy - 2yz + 4xz\)

iii)\((-2x + 3y + 2z)^2 = (-2x)^2 + (3y)^2 +(2z)^2 - 2(2x)(3y) + 2(3y)(2z) - 2(2x)(2z)\)
by using identity (v)
i.e., \((a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ac\)
\(= 4x^2 + 9y^2 + 4z^2 - 12xy + 12yz - 8xz\)

iv)\((3a - 7b - c)^2 = (3a)^2 + (-7b)^2 + (-c)^2 - 2(3a)(7b) + 2(-7b)(-c) - 2(3a)(c)\)
by using identity (v)
i.e., \((a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ac\)
\(= 9a^2 + 49b^2 + c^2 - 42ab + 14bc - 6ac\)

v)\((-2x + 5y - 3z)^2 = (-2x)^2 + (5y)^2 +(-3z)^2 - 2(2x)(5y) - 2(5y)(3z) + 2(2x)(3z)\)
by using identity (v)
i.e., \((a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ac\)
\(= 4x^2 + 25y^2 + 9z^2 - 20xy - 30yz + 12xz\)

vi)\([1/4a - 1/2b + 1]^2 = (1/4a)^2 + (-1/2b)^2 + (1)^2 - 2(1/4a)(1/2b) - 2(-1/2b)(1) + 2(1/4a)(1)\)
by using identity (v)
i.e., \((a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ac\)
\(= a^2/16 + b^2/4 + 1 - ab/4 - b + a/2\)