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If-P-x-aX-3-bX-c-Q-X-a-0-has-the-roots-x-1-x-2-and-x-3-such-that-x-1-x-2-x-3-then-prove-that-x-1-0-if-and-only-if-b-c-




Question Number 153161 by mathdanisur last updated on 05/Sep/21
If  P(x)=aX^3 +bX+c ∈ Q[X]  ;  (a≠0)  has the roots  x_1  , x_2   and  x_3   such that  x_1  = x_2 x_3   then prove that x_1  = 0 if and only if b=c
$$\mathrm{If}\:\:\mathrm{P}\left(\mathrm{x}\right)=\mathrm{aX}^{\mathrm{3}} +\mathrm{bX}+{c}\:\in\:\mathrm{Q}\left[\mathrm{X}\right]\:\:;\:\:\left(\mathrm{a}\neq\mathrm{0}\right) \\ $$$$\mathrm{has}\:\mathrm{the}\:\mathrm{roots}\:\:\mathrm{x}_{\mathrm{1}} \:,\:\mathrm{x}_{\mathrm{2}} \:\:\mathrm{and}\:\:\mathrm{x}_{\mathrm{3}} \\ $$$$\mathrm{such}\:\mathrm{that}\:\:\mathrm{x}_{\mathrm{1}} \:=\:\mathrm{x}_{\mathrm{2}} \mathrm{x}_{\mathrm{3}} \\ $$$$\mathrm{then}\:\mathrm{prove}\:\mathrm{that}\:\mathrm{x}_{\mathrm{1}} \:=\:\mathrm{0}\:\mathrm{if}\:\mathrm{and}\:\mathrm{only}\:\mathrm{if}\:\mathrm{b}=\mathrm{c} \\ $$
Answered by mr W last updated on 05/Sep/21
x_1 +x_2 +x_3 =0  x_1 (x_2 +x_3 )+x_2 x_3 =(b/a)  x_1 x_2 x_3 =−(c/a)    with x_1 =x_2 x_3 :  −x_1 ^2 +x_1 =(b/a)  x_1 ^2 =−(c/a)  ⇒x_1 =((b−c)/a)  x_1 =0 if and only if b−x=0 ⇒b=c
$${x}_{\mathrm{1}} +{x}_{\mathrm{2}} +{x}_{\mathrm{3}} =\mathrm{0} \\ $$$${x}_{\mathrm{1}} \left({x}_{\mathrm{2}} +{x}_{\mathrm{3}} \right)+{x}_{\mathrm{2}} {x}_{\mathrm{3}} =\frac{{b}}{{a}} \\ $$$${x}_{\mathrm{1}} {x}_{\mathrm{2}} {x}_{\mathrm{3}} =−\frac{{c}}{{a}} \\ $$$$ \\ $$$${with}\:{x}_{\mathrm{1}} ={x}_{\mathrm{2}} {x}_{\mathrm{3}} : \\ $$$$−{x}_{\mathrm{1}} ^{\mathrm{2}} +{x}_{\mathrm{1}} =\frac{{b}}{{a}} \\ $$$${x}_{\mathrm{1}} ^{\mathrm{2}} =−\frac{{c}}{{a}} \\ $$$$\Rightarrow{x}_{\mathrm{1}} =\frac{{b}−{c}}{{a}} \\ $$$${x}_{\mathrm{1}} =\mathrm{0}\:{if}\:{and}\:{only}\:{if}\:{b}−{x}=\mathrm{0}\:\Rightarrow{b}={c} \\ $$
Commented by mathdanisur last updated on 05/Sep/21
veri nice thank you ser
$$\mathrm{veri}\:\mathrm{nice}\:\mathrm{thank}\:\mathrm{you}\:\mathrm{ser} \\ $$

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