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Question Number 35256 by bivekverma last updated on 17/May/18

Factorize :x^5 −y^5

$$\mathrm{Factorize}\::\mathrm{x}^{\mathrm{5}} −\mathrm{y}^{\mathrm{5}} \\ $$

Commented by abdo mathsup 649 cc last updated on 18/May/18

let factorize x^n  −y^n   for n integr  letsuppose y≠o  x^n  −y^n =y^n  (  ((x/y))^n  −1)  =y^n (  (x/y)−1)( 1  +(x/y) +(x^2 /y^2 ) +....(x^(n−1) /y^(n−1) ))  =y^(n−1) (x−y)( 1+ (x/y) +(x^2 /y^2 ) + ....(x^(n−1) /y^(n−1) ))  =(x−y)( y^(n−1)   +xy^(n−2)  +x^2 y^(n−3)  +...+x^(n−1) ) for  n=5 we get  x^5  −y^5 =(x−y)(y^4  +xy^3  +x^2  y^2  +x^3 y +x^4 )  =(x−y)(x^4  +x^3 y +x^2 y^2   +xy^3  +y^4 )  we see that x^n −y^n =(x−y)Σ_(i+j=n−1) x^i y^j

$${let}\:{factorize}\:{x}^{{n}} \:−{y}^{{n}} \:\:{for}\:{n}\:{integr}\:\:{letsuppose}\:{y}\neq{o} \\ $$$${x}^{{n}} \:−{y}^{{n}} ={y}^{{n}} \:\left(\:\:\left(\frac{{x}}{{y}}\right)^{{n}} \:−\mathrm{1}\right) \\ $$$$={y}^{{n}} \left(\:\:\frac{{x}}{{y}}−\mathrm{1}\right)\left(\:\mathrm{1}\:\:+\frac{{x}}{{y}}\:+\frac{{x}^{\mathrm{2}} }{{y}^{\mathrm{2}} }\:+....\frac{{x}^{{n}−\mathrm{1}} }{{y}^{{n}−\mathrm{1}} }\right) \\ $$$$={y}^{{n}−\mathrm{1}} \left({x}−{y}\right)\left(\:\mathrm{1}+\:\frac{{x}}{{y}}\:+\frac{{x}^{\mathrm{2}} }{{y}^{\mathrm{2}} }\:+\:....\frac{{x}^{{n}−\mathrm{1}} }{{y}^{{n}−\mathrm{1}} }\right) \\ $$$$=\left({x}−{y}\right)\left(\:{y}^{{n}−\mathrm{1}} \:\:+{xy}^{{n}−\mathrm{2}} \:+{x}^{\mathrm{2}} {y}^{{n}−\mathrm{3}} \:+...+{x}^{{n}−\mathrm{1}} \right)\:{for} \\ $$$${n}=\mathrm{5}\:{we}\:{get} \\ $$$${x}^{\mathrm{5}} \:−{y}^{\mathrm{5}} =\left({x}−{y}\right)\left({y}^{\mathrm{4}} \:+{xy}^{\mathrm{3}} \:+{x}^{\mathrm{2}} \:{y}^{\mathrm{2}} \:+{x}^{\mathrm{3}} {y}\:+{x}^{\mathrm{4}} \right) \\ $$$$=\left({x}−{y}\right)\left({x}^{\mathrm{4}} \:+{x}^{\mathrm{3}} {y}\:+{x}^{\mathrm{2}} {y}^{\mathrm{2}} \:\:+{xy}^{\mathrm{3}} \:+{y}^{\mathrm{4}} \right) \\ $$$${we}\:{see}\:{that}\:{x}^{{n}} −{y}^{{n}} =\left({x}−{y}\right)\sum_{{i}+{j}={n}−\mathrm{1}} {x}^{{i}} {y}^{{j}} \\ $$

Answered by MJS last updated on 17/May/18

x^5 −y^5 =(x−y)(x^4 +x^3 y+x^2 y^2 +xy^3 +y^4 )

$${x}^{\mathrm{5}} −{y}^{\mathrm{5}} =\left({x}−{y}\right)\left({x}^{\mathrm{4}} +{x}^{\mathrm{3}} {y}+{x}^{\mathrm{2}} {y}^{\mathrm{2}} +{xy}^{\mathrm{3}} +{y}^{\mathrm{4}} \right) \\ $$

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