E:7042

E:7042 (Mariș Ilieș)

Fie 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 F = \frac{x^4 - 10x^3 + 35x^2 - 50x +24}{x^3 - 9x^2 + 26x - 24}} . Să se determine 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 \in \mathbb{R}} pentru care 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 F} are sens și să se simpifice această fracție.

Soluție.

Avem 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^3 - 9x^2 + 26x - 24 = x^3 - 2x^2 - 7x^2 + 14x + 12x-24=\left(x-2\right) \left(x^2-7x+12\right) = \left(x-2\right)\left(x-3\right)\left(x-4\right)} . Deci, fracția 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 F} este bine definită pentru orice 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\ in \mathbb{R}\setminus\{2,3,4\}} .

Avem $x^{4}-10x^{3}+35x^{2}-50x+24=x^{4}-x^{3}-9x^{3}+9x^{2}+26x^{2}-26x-24x+24=(x-1)\left(x^{3}-9x^{2}+26x-24\right)$. Rezultă 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 F = x-1} .