E:15651

E:15651 (Mihai Pălincaș, elev)

Determinați soluțiile raționale ale ecuației Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^{2020} - x^2 - 3x = 0} .

Soluție

Ecuația se scrie Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x\left(x^{2019} - x - 3 \right) = 0} , de unde $x=0$ sau $x^{2019}-x-3=0$.

Vom demonstra arăta că ecuația $x^{2019}-x-3=0$ nu are soluții raționale. Presupunem că există o soluție rațională ${\frac {p}{q}}$, cu $p$ și $q$ numere întregi, prime între ele. Atunci Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{p^{2019}}{q^{2019}} - \frac{p}{q} -3 = 0} , ceea ce revine la Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p^{2019} - p \cdot q^{2018} - 3\cdot q^{2019} = 0} . Cum Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q | p\cdot q^{2018}} și Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q | q^{2019}} , rezultă că Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q | p^{2019}} , ceea ce este în contradicție cu Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left(p,q\right) = 1} . Deci, ecuația Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^{2019} - x - 3 = 0} nu are soluții raționale.

În concluzie, singura soluție rațională a ecuației Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^{2020} - x^2 - 3x = 0} este Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x = 0} .