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8.Factorize each of the following :
i)\(8a^3 + b^3 + 12a^2b + 6ab^2\)
ii)\(8a^3 - b^3 - 12a^2b + 6ab^2\)
iii)\(27 - 125a^3 - 135a + 225a^2\)
iv)\(64a^3 - 27b^3 - 144a^2b + 108ab^2\)
v)\(27p^3 - 1/216 - 9p^2/2 + p/4\)
Answer :

i)\(8a^3 + b^3 + 12a^2b + 6ab^2 \)
i.e.,\(= (2a)^3 + b^3 +3(2a)(b)(2a + b))\)
by using identity \((a + b)^3 = a^3 + b^3 + 3ab(a + b)\)
\(= (2a)^3 + b^3\)

ii)\(8a^3 - b^3 - 12a^2b + 6ab^2 \)
i.e.,\(= (2a)^3 - b^3 -3(2a)(b)(2a - b)\)
by using identity \((a - b)^3 = a^3 - b^3 - 3ab(a - b)\)
\(= (2a)^3 - b^3\)

iii)\(27 - 125a^3 - 135a + 225a^2 \)
i.e.,\(= (3)^3 - (5a)^3 -3(3)(5a)(3 - 5a)\)
by using identity \((a - b)^3 = a^3 - b^3 - 3ab(a - b)\)
\(= (3)^3 - 5a^3\)

iv)\(64a^3 - 27b^3 - 144a^2b + 108ab^2\)
i.e.,\( = (4a)^3 - (3b)^3 -3(4a)(4b)(4a - 3b))\)
by using identity \((a - b)^3 = a^3 - b^3 - 3ab(a - b)\)
\(= (4a)^3 - (3b)^3\)

v)\((27p^3 - 1/216 - 9p^2/2 + p/4\)
i.e.,\( = (3p)^3 - (1/6)^3 -3(3p)(1/6)(3p - 1/6))\)
by using identity \((a - b)^3 = a^3 - b^3 - 3ab(a - b)\)
\(= (3p)^3 - {1/6}^3\)