Q. यदि `x+y+z=6` , `xyz= -10` तथा `x^2+y^2+z^2 = 30` है , तो `x^3 +y^3+z^3` का मान क्या होगा ?

Q. यदि $x+y+z=6$ , $xyz= -10$ तथा $x^2+y^2+z^2 = 30$ है , तो $x^3 +y^3+z^3$ का मान क्या होगा ?

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$(x^3+y^3+z^3)-3xyz = (x+y+z) ( x^2+y^2+z^2-xy-yz-zx)$

$(x^3+y^3+z^3) = (x+y+z) ( x^2+y^2+z^2-xy-yz-zx)+3xyz$

first of all we have to find $xy-yz-zx$

$x+y+z=6$

On Squaring bothsides we have

$(x+y+z)^2 = ( x^2+y^2+z^2 + 2xy +2yz+2zx )$

$(6)^2 = 30 + 2 ( xy + yz +zx )$

$36 = 30 + 2 ( xy + yz + zx )$

$36-.30 = 2 ( xy + yz + zx )$

$6 = 2 ( xy + yz + zx )$

$( xy + yz + zx )= 3$

Now,

$(x^3+y^3+z^3) = (x+y+z) ( x^2+y^2+z^2-xy-yz-zx)+3xyz$

$(x^3+y^3+z^3) = 6\times (30-3) + 3\times -10$

$(x^3+y^3+z^3)= 6 \times27 -30$

$(x^3+y^3+z^3) = 162-30$

$(x^3+y^3+z^3)= 132$

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Q. यदि $x+y+z=6$ , $xyz= -10$ तथा $x^2+y^2+z^2 = 30$ है , तो $x^3 +y^3+z^3$ का मान क्या होगा ?