गुणनखण्ड कीजिए : (a) `x^(6)+y^(6)` (b) `x^(6)-y^(6)`

Explanation

(a) यहाँ
`x^(6)+y^(6)=(x^(2))^(3)+(y^(2))^(3)`
`=(x^(2)+y^(2))(x^(4)-x^(2)y^(2)+y^(4))` . . .(1)
`[becausea^(3)+b^(3)=(a+b)(a^(2)-ab+b^(2))]`
ज्यादातर यहाँ तक गुणनखण्ड करते है जो सही नहीं है । इसके आगे गुणनखण्ड होंगे
`x^(6)+y^(6)=(x^(2)+y^(2))[underset("पूर्ण वर्ग बनाने पर")underbrace((x^(2))^(2)+2x^(2)*y^(2)+(y^(2))^(2))-2x^(2)y^(2)-x^(2)y^(2)]`
`rArr" "x^(6)+y^(6)=(x^(2)+y^(2))[(x^(2)+y^(2))^(2)-3x^(2)y^(2)]=(x^(2)+y^(2))[(x^(2)+y^(2))^(2)-(sqrt(3)xy)^(2)]`
`=(x^(2)+y^(2))[(x^(2)+y^(2)+sqrt(3)xy)(x^(2)+y^(2)-sqrt(3)xy)]`
`rArr" "underset("बहुपद")underbrace(x^(6)+y^(6))=underset("बहुपद")underbrace((x^(2)+y^(2)))underset("बहुपद")underbrace((x^(2)+sqrt(3)xy+y^(2)))underset("बहुपद")underbrace((x^(2)-sqrt(3)xy+y^(2)))`
(b) `x^(6)-y^(6)=(x^(2))^(3)-(y^(2))^(3)=(x^(2)-y^(2))(x^(4)+x^(2)y^(2)+y^(4))`
`=(x+y)(x-y)[underset("पूर्ण वर्ग बनाने पर")underbrace((x^(2))^(2)+2(x^(2))(y^(2))+(y^(2))^(2))-2x^(2)y^(2)+x^(2)y^(2)]`
`=(x+y)(x-y)[(x^(2)+y^(2))^(2)-(xy)^(2)]`
`=(x+y)(x-y)[(x^(2)+y^(2)+xy)(x^(2)+y^(2)-xy)]`
`=(x+y)(x-y)(x^(2)+xy+y^(2))(x^(2)-xy+y^(2))`
अन्य विधि :
`x^(6)-y^(6)=(x^(3))^(2)-(y^(3))^(2)=(x^(3)+y^(3))(x^(3)-y^(3))`
`=(x+y)(x^(2)-xy+y^(2))(x-y)(x^(2)+xy+y^(2))`
`=(x+y)(x-y)(x^(2)+xy+y^(2))(x^(2)-xy+y^(2))`