# Multiply:

Question:

Multiply:

(i) $x^{2}+y^{2}+z^{2}-x y+x z+y z$ by $x+y-z$

(ii) $x^{2}+4 y^{2}+z^{2}+2 x y+x z-2 y z b y x-2 y-z$

(iii) $x^{2}+4 y^{2}+2 x y-3 x+6 y+9$ by $(x-2 y+3)$

(iv) $9 x^{2}+25 y^{2}+15 x y+12 x-20 y+16$ by $3 x-5 y+4$

Solution:

(i) $x^{2}+y^{2}+z^{2}-x y+x z+y z$ by $x+y-z$

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

$=x^{3}+y^{3}+z^{3}-3 x y z$

(ii) $x^{2}+4 y^{2}+z^{2}+2 x y+x z-2 y z b y x-2 y-z$

$x^{2}+(-2 y)^{2}+(-z)^{2}-(-2 y)(-z)-(-z)(x)$

$=x^{3}+(-2 y)^{3}+(-z)^{3}-3 x(-2 y)(-z)$

$\Rightarrow x^{2}+4 y^{2}+z^{2}+2 x y-2 y z+z x$

$=x^{3}-8 y^{3}-z^{3}-6 x y z$

(iii) $x^{2}+4 y^{2}+2 x y-3 x+6 y+9$ by $(x-2 y+3)$

$(x)^{2}+(-2 y)^{2}+(3)^{2}-(x)(-2 y)-(-2 y)(3)-3(x)$

$=(x)^{3}+(-2 y)^{3}+3^{3}-3(x)(-2 y)(3)$

$\Rightarrow x^{2}+4 y^{2}+9+2 x y+6 y-3 x$

$=x^{3}-8 y^{3}+27+18 x y$

(iv) $9 x^{2}+25 y^{2}+15 x y+12 x-20 y+16$ by $3 x-5 y+4$

$(3 x)^{2}+(5 y)^{2}+4^{2}-(-3 x)(5 y)-(5 y)(4)-(4)(-3 x)$

$=(-3 x)^{3}+(5 y)^{3}+4^{3}-3(-3 x)(5 y)(4)$

$\Rightarrow 9 x^{2}+25 y^{2}+16+15 x y-20 y+12 x$

$=-27 x^{3}+125 y^{3}+64+180 x y$