Question

Factors of $27x^3+y^3+z^3-9xyz$ are:

• A. $(3x+y+z)(9x^2+y^2+z^2-3xy-yz-3xz)$
• B. $(3x+y+z)(9x^2+y^2+z^2+3xy+yz+3xz)$
• C. $(3x-y+z)(9x^2+y^2-z^2+3xy+yz+3xz)$
• D. $(3x-y-z)(9x^2+y^2+z^2-3xy-yz-3xz)$

Answer 1

The correct option is A

$(3x+y+z)(9x^2+y^2+z^2-3xy-yz-3xz)$

$27x^3+y^3+z^3-9xyz$
$=(3x{)}^3+(y{)}^3+(z{)}^3-3(3x\left)\right(y\left)\right(z)$
We know, $a^3+b^3+c^3-3abc=(a+b+c)(a^2+b^2+c^2-ab-bc-ca)$
$\Rightarrow (3x{)}^3+(y{)}^3+(z{)}^3-3(3x\left)\right(y\left)\right(z)$
$=(3x+y+z)\left[\right(3x{)}^2+(y{)}^2+(z{)}^2-\left(3x\right)\left(y\right)-\left(y\right)\left(z\right)-\left(3x\right)z]$
$\Rightarrow 27x^3+y^3+z^3-9xyz$
$=(3x+y+z)(9x^2+y^2+z^2-3xy-yz-3xz)$