Ex.2.5 Q12 Polynomials Solution - NCERT Maths Class 9

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Question

Verify that:

\[\begin{align}x^{3}+y^{3}+z^{3}-3 x y=\frac{1}{2}(x+y+z)\left[(x-y)^{2}+(y-z)^{2}+(z-x)^{2}\right]\end{align}\]

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Polynomials
Ex 2.5 | Question 12

Text Solution

  

Reasoning:

Identity:  \(\begin{align}x^{3}+y^{3}+z^{3}-3 x y z=(x+y+z)\left(x^{2}+y^{2}+z^{2}-x y-y z-z x\right)\end{align}\)

Steps:

\[\begin{align} \text { Taking RHS } &=\frac{1}{2}(x+y+z)\left[(x-y)^{2}+(y-z)^{2}+(z-x)^{2}\right] \\ &=\frac{1}{2}(x+y+z)\left[\left(x^{2}-2 x y+y^{2}+y^{2}-2 y z+z^{2}+z^{2}-2 z x+x^{2}\right]\right.\\ &=\frac{1}{2}(x+y+z)\left[2 x^{2}+2 y^{2}+2 z^{2}-2 x y-2 y z-2 z x\right] \\ &=\frac{1}{2}(x+y+z)(2)\left[x^{2}+y^{2}+z^{2}-x y-y z-z x\right]\\&=x\left[x^{2}+y^{2}+z^{2}-x y-y z-z x\right]\\&\qquad+y\left[x^{2}+y^{2}+z^{2}-x y-y z-z x\right]\\&\qquad+z\left[x^{2}+y^{2}+z^{2}-x y-y z-z x\right]\\ &=\left[x^{3}+x y^{2}+x z^{2}-x^{2} y-x y z-x^{2} z+x^{2} y\right.\\&\qquad +y^{3}+y z^{2}-x y^{2}-y^{2} z-x y z+z x^{2}+y^{2} z+z^{3}-x y z-y z^{2}-x z^{2} ] \\ &=x^{3}+y^{3}+z^{3}-3 x y z=\mathrm{LHS}\end{align}\]