Ex.2.5 Q9 Polynomials Solution - NCERT Maths Class 9

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Question

Verify:

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

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

   

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Polynomials
Ex 2.5 | Question 9

Text Solution

  

Steps:

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

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

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

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