Question

factorise the equation:1+x^5

Answer 1

Hey

Put : x = ( -1 ) to get :
$(1 + {x}^{5} ) = (1 + ( - 1)) = 0$

Hence, by Factor theorem, ( 1 + x ) is always a factor.

$= > p(x) = ( 1 + x)(1 - x + {x}^{2} - {x}^{3} + {x}^{4} )$
The other factor is a Cyclotomic Polynomial and hence, cannot be factorized over real numbers

Hence, the required factorization is :
$(1 + {x}^{5} ) = (1 + x)(1 - x + {x}^{2} - {x}^{3} + {x}^{4} )$