if x^5 +x+1=0, find x^3 −x^2 =?


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Question Number 195809 by mr W last updated on 11/Aug/23

$i f x^{5} + x + 1 = 0, f i n d x^{3} - x^{2} = ?$
Commented by jabarsing last updated on 11/Aug/23

$h e l l o d e a r W,$ $1 ?$
Commented by jabarsing last updated on 11/Aug/23

$i f t h a t i s m y q u e s t i o n; I w a n a x^{3} - x^{2} o r$ $x^{2} - x^{3}$
Commented by mr W last updated on 11/Aug/23

$y e s s i r . i m e a n t x^{3} - x^{2} i n d e e d .$
	Answered by MM42 last updated on 11/Aug/23

	$x^{5} + x + 1 = (x^{3} - x^{2} + 1) (x^{2} + x + 1) = 0$ $\overset{i f x \in R}{\Rightarrow} x^{3} - x^{2} = - 1 ✓$
	Commented by mr W last updated on 11/Aug/23

	$g r e a t!$
	Answered by manxsol last updated on 12/Aug/23

	$x^{5} + x + 1 = 0 \Rightarrow x^{3} - x^{2} = ?$ $- - - - - - - - - - - -$ $E = x^{3} - x^{2}$ $x^{3} = E + x^{2}$ $x^{5} = {E x}^{2} + x . x^{3}$ $x^{5} = {E x}^{2} + x (E + x^{2})$ $x^{5} = {E x}^{2} + x E + E + x^{2}$ $x^{5} + x + 1 = E (x^{2} + x + 1) + x^{2} + x + 1$ $0 = (x^{2} + x + 1) (E + 1)$ $E = - 1$
	Commented by mr W last updated on 12/Aug/23
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