Factorize-x-5-y-5-

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

Factorize : x^{5} - y^{5}

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

l e t f a c t o r i z e x^{n} - y^{n} f o r n i n t e g r l e t s u p p o s e y \neq o

x^{n} - y^{n} = y^{n} ({(\frac{x}{y})}^{n} - 1)

= y^{n} (\frac{x}{y} - 1) (1 + \frac{x}{y} + \frac{x^{2}}{y^{2}} + \dots . \frac{x^{n - 1}}{y^{n - 1}})

= y^{n - 1} (x - y) (1 + \frac{x}{y} + \frac{x^{2}}{y^{2}} + \dots . \frac{x^{n - 1}}{y^{n - 1}})

= (x - y) (y^{n - 1} + {x y}^{n - 2} + x^{2} y^{n - 3} + \dots + x^{n - 1}) f o r

n = 5 w e g e t

x^{5} - y^{5} = (x - y) (y^{4} + {x y}^{3} + x^{2} y^{2} + x^{3} y + x^{4})

= (x - y) (x^{4} + x^{3} y + x^{2} y^{2} + {x y}^{3} + y^{4})

w e s e e t h a t x^{n} - y^{n} = (x - y) \sum_{i + j = n - 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} + {x y}^{3} + y^{4})

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