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2. Polynomials
hard
જો $x+y+z=0,$ તો સાબિત કરો કે $x^{3}+y^{3}+z^{3}=3 x y z$.
Option A
Option B
Option C
Option D
Solution
$x+y+z=0 $
$\therefore x+y=-z$
$\therefore $ $(x+y)^{3}=(-z)^{3}$ ($\because $ બંને બાજુ ઘન લેતા)
$\therefore $ $x^{3}+y^{3}+3 x y(-z)=-z^{3}$
$\therefore $ $\left(x^{3}+y^{3}+z^{3}\right)-3 x y z=0$
$\therefore $ $\left(x^{3}+y^{3}+z^{3}\right)=3 x y z$
આમ, $x+y+z=0,$ હોય તો $\left(x^{3}+y^{3}+z^{3}\right)=3 x y z$
Standard 9
Mathematics
