Let t be a root of x^3 −3x+1=0, if ((t^2 +pt+1)/(t^2 −t+1)) can be written as t+c for some p,c ∈ Z, then p−c equals?


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Question Number 110419 by Aina Samuel Temidayo last updated on 28/Aug/20

$Let t be a root of x^{3} - 3 x + 1 = 0, if$ $\frac{t^{2} + pt + 1}{t^{2} - t + 1} can be written as t + c for$ $some p, c \in Z, then p - c equals ?$
Commented by Her_Majesty last updated on 28/Aug/20

$p, z \in Z o r p, c \in Z ?$
Commented by Aina Samuel Temidayo last updated on 28/Aug/20

$Oh . Sorry for that . It has been$ $corrected .$
	Answered by Her_Majesty last updated on 28/Aug/20

	$t^{2} + p t + 1 = (t + c) (t^{2} - t + 1)$ $\Leftrightarrow$ $t^{3} + (c - 2) t^{2} - (c + p - 1) t + c - 1 = 0$ $b u t a l s o x = t i n 1^{s t} e q u a t i o n$ $t^{3} - 3 t + 1 = 0$ $\Rightarrow$ $(1) c - 2 = 0$ $(2) c + p - 1 = 3$ $(3) c - 1 = 1$ $\Rightarrow c = 2 \land p = 2$
	Commented by Aina Samuel Temidayo last updated on 28/Aug/20

	$Implies p - c = 0 right ?$
	Commented by Aina Samuel Temidayo last updated on 28/Aug/20

	$Why are we dealing with the$ $numerator only ? Could you please$ $shed more light on it ? And why did$ $you equate the numerator to 0 ?$
	Commented by Her_Majesty last updated on 28/Aug/20

	$‘ ‘ \frac{t^{2} + pt + 1}{t^{2} - t + 1} can be written as t + c^{″}$ $m e a n s$ $\frac{t^{2} + pt + 1}{t^{2} - t + 1} = t + c \Leftrightarrow t^{2} + pt + 1 = (t + c) (t^{2} - t + 1)$ $t h i s i s w h a t I d i d, t h e r e s t f o l l o w s$
	Commented by Aina Samuel Temidayo last updated on 28/Aug/20

	$Thanks, I really appreciate that . You$ $can also look up other questions I$ $posted .$
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