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Ex.2.5 Q13 Polynomials Solution - NCERT Maths Class 9

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Question

If  \(\begin{align}x+y+z=0,\end{align}\)

show that  \(\begin{align}x^{3}+y^{3}+z^{3}=3 x y z\end{align}\)

   

 Video Solution
Polynomials
Ex 2.5 | Question 13

Text Solution

  

Reasoning:

 Identity:

\(\begin{align}&x^{3}+y^{3}+z^{3}-3 x y z\\&=\!(x\!+\!y\!+\!z)\left(x^{2}\!+\!y^{2}\!+\!z^{2}\!-\!x y\!-\!y z\!-\!zx\right)\end{align}\)

Steps:

By the identity:

\(\begin{align}&x^{3}+y^{3}+z^{3}-3 x y z\\&=\!(x\!+\!y\!+\!z)\left(x^{2}\!+\!y^{2}\!+\!z^{2}\!-\!x y\!-\!y z\!-\!zx\right)\end{align}\)

If  \(x + y + z = 0\)  then the entire \(RHS\) becomes \(0\) and hence the \(LHS\)

\(\begin{align}x^{3}+y^{3}+z^{3}-3 x y z=0\end{align}\)

Hence,

\[\begin{align}x^{3}+y^{3}+z^{3}=3 x y z\end{align}\]

 Video Solution
Polynomials
Ex 2.5 | Question 13