Resolve each of the following quadratic trinomial into factor:
15x2 − 16xyz − 15y2z2
The given expression is $15 x^{2}-16 x y z-15 y^{2} z^{2}$. (Coefficient of $x^{2}=15$, coefficient of $x=-16 y z$ and constant term $=-15 y^{2} z^{2}$ )
Now, we will split the coefficient of $x$ into two parts such that their sum is $-16 y z$ and their product equals the product of the coefficient of $x^{2}$ and the cons $\tan t$ term, i.e., $15 \times\left(-15 y^{2} z^{2}\right)=-225 y^{2} z^{2}$.
Now,
$(-25 y z)+9 y z=-16 y x$
and
$(-25 y z) \times 9 y z=-225 y^{2} z^{2}$
Replacing the middle term $-16 x y z$ by $-25 x y z+9 x y z$, we have :
$15 x^{2}-16 x y z-15 y^{2} z^{2}=15 x^{2}-25 x y z+9 x y z-15 y^{2} z^{2}$
$=\left(15 x^{2}-25 x y z\right)+\left(9 x y z-15 y^{2} z^{2}\right)$
$=5 x(3 x-5 y z)+3 y z(3 x-5 y z)$
$=(5 x+3 y z)(3 x-5 y z)$
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