Question

Q.11 Factorize the following by using a suitable identity: (a) a3 +b3 - 8c3 + 6abc (b) 8x3 -27y3 + 125 z3 + 90xyz Q.12 Find the value of a3 + 8b3 if a + 2b = 10 and ab =15.b​

Answer 1

Answer:

We know the identity

$a {}^{3} + b {}^{3} + c ^{3} - 3abc = (a + b + c)(a^{2} + b^{2} + c^{2} - ab - bc - ca$

Using the above identity taking a=a,b=−b and c=2c, the equation

a^3−b^3+8c^3+6abc can be factorised as follows:

a^3−b^3+8c^3+6abc=(a^3)+(−b)^3+(2c)^3−3(a)(−b)(2c),

=[a+(−b)+(2c)][a^2+(−b)^2+(2c)^2−(a×−b)−(−b×2c)−(2c×a)],

=(a−b+2c)(a^2+b^2+4c^2+ab+2bc−2ca),

Hence, a^3−b^3+8c^3+6abc=(a−b+2c)(a^2+b^2+4c^2+ab+2bc−2ca).