ঘন-এর সাহায্যে সরল কর: $(2x+3y-4z)^3$$+(2x-3y+4z)^3$$+12x\left\{4x^2-(3y-4z)^2\right\}$

মনে করি,
$(2x+3y-4z)=p$

এবং, $(2x-3y+4z)=q$

সুতরাং,
$p\times q$

$(2x+3y-4z)(2x-3y+4z)$

$=\left\lbrace2x+\left(3y-4z\right)\right\rbrace\left\lbrace2x-\left(3y-4z\right)\right\rbrace$

$=\left\lbrace\left(2x\right)^2-\left(3y-4z\right)^2\right\rbrace$

$=\left\lbrace4x^2-\left(3y-4z\right)^2\right\rbrace$

আবার,
$p+q$

$=(2x+3y-4z)+(2x-3y+4z)$

$=2x+3y-4z+2x-3y+4z$

$=4x$

প্রদত্ত রাশি,
$(2x+3y-4z)^3$$+(2x-3y+4z)^3$$+12x\left\{4x^2-(3y-4z)^2\right\}$

$=(2x+3y-4z)^3$$+(2x-3y+4z)^3$$+3\times4x\times\left\{4x^2-(3y-4z)^2\right\}$

$=(2x+3y-4z)^3$$+(2x-3y+4z)^3$$+3\times\left\{4x^2-(3y-4z)^2\right\}\times4x$

$=p^3+q^3+3\times pq\times\left(p+q\right)$
[ মান বসিয়ে ]

$=p^3+q^3+3pq\left(p+q\right)$

$=\left(p+q\right)^3$

$=\left(4x\right)^3$
[ মান বসিয়ে ]

$=64x^3$ [Answer]