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Question Number 173196 by AgniMath last updated on 08/Jul/22
What is the value of  ((y^2  + yz + z^2 )/((x − y)(x − z)))+((z^2  + zx + x^2 )/((y − z)(z − y)))+((x^2  + xy + y^2 )/((z − x)(z−y)))
$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of} \\ $$$$\frac{{y}^{\mathrm{2}} \:+\:{yz}\:+\:{z}^{\mathrm{2}} }{\left({x}\:−\:{y}\right)\left({x}\:−\:{z}\right)}+\frac{{z}^{\mathrm{2}} \:+\:{zx}\:+\:{x}^{\mathrm{2}} }{\left({y}\:−\:{z}\right)\left({z}\:−\:{y}\right)}+\frac{{x}^{\mathrm{2}} \:+\:{xy}\:+\:{y}^{\mathrm{2}} }{\left({z}\:−\:{x}\right)\left({z}−{y}\right)} \\ $$
Commented by som(math1967) last updated on 08/Jul/22
0
$$\mathrm{0} \\ $$
Answered by Rasheed.Sindhi last updated on 08/Jul/22
((y^2  + yz + z^2 )/((x − y)(x − z)))+((z^2  + zx + x^2 )/((y − z)(z − y)))+((x^2  + xy + y^2 )/((z − x)(z−y)))  (((y−z)(y^2  + yz + z^2 )+(z−x)(z^2  + zx + x^2 )+(x−y)(x^2  + xy + y^2 ))/((x−y)(y−z)(z−x)))  (((y^3 −z^3 )+(z^3 −x^3 )+(x^3 −y^3 ))/((x−y)(y−z)(z−x)))  (0/((x−y)(y−z)(z−x)))=0
$$\frac{{y}^{\mathrm{2}} \:+\:{yz}\:+\:{z}^{\mathrm{2}} }{\left({x}\:−\:{y}\right)\left({x}\:−\:{z}\right)}+\frac{{z}^{\mathrm{2}} \:+\:{zx}\:+\:{x}^{\mathrm{2}} }{\left({y}\:−\:{z}\right)\left({z}\:−\:{y}\right)}+\frac{{x}^{\mathrm{2}} \:+\:{xy}\:+\:{y}^{\mathrm{2}} }{\left({z}\:−\:{x}\right)\left({z}−{y}\right)} \\ $$$$\frac{\left({y}−{z}\right)\left({y}^{\mathrm{2}} \:+\:{yz}\:+\:{z}^{\mathrm{2}} \right)+\left({z}−{x}\right)\left({z}^{\mathrm{2}} \:+\:{zx}\:+\:{x}^{\mathrm{2}} \right)+\left({x}−{y}\right)\left({x}^{\mathrm{2}} \:+\:{xy}\:+\:{y}^{\mathrm{2}} \right)}{\left({x}−{y}\right)\left({y}−{z}\right)\left({z}−{x}\right)} \\ $$$$\frac{\left({y}^{\mathrm{3}} −{z}^{\mathrm{3}} \right)+\left({z}^{\mathrm{3}} −{x}^{\mathrm{3}} \right)+\left({x}^{\mathrm{3}} −{y}^{\mathrm{3}} \right)}{\left({x}−{y}\right)\left({y}−{z}\right)\left({z}−{x}\right)} \\ $$$$\frac{\mathrm{0}}{\left({x}−{y}\right)\left({y}−{z}\right)\left({z}−{x}\right)}=\mathrm{0} \\ $$

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