Question

factorize the expression:-
x^5+x+1

Answer 1

$x {}^{5} + x + 1 = x {}^{5} + x + 1 + x {}^{2} - x {}^{2} \\ = x {}^{5} - x {}^{2} + x {}^{2} + x + 1 \\ = ( x {}^{5} - x {}^{2} ) + (x {}^{2} + x + 1) \\ = x {}^{2} (x {}^{3} - 1) + (x {}^{2} + x + 1) \\ = x {}^{2} (x - 1)(x {}^{2} + x + 1) + (x {}^{2} + x + 1) \\ = (x {}^{2} + x + 1)(x {}^{2} (x - 1) + 1) \\ = (x {}^{2} + x + 1)(x {}^{3} - x {}^{2} + 1)$

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