\(\displaystyle{ x^{4} - 4 x^{3} + 1 = 0}\)
Jak to rozwiązać?
Wielomian czwartego stopnia
- Mariusz M
- Użytkownik
- Posty: 6903
- Rejestracja: 25 wrz 2007, o 01:03
- Płeć: Mężczyzna
- Lokalizacja: 53°02'N 18°35'E
- Podziękował: 2 razy
- Pomógł: 1246 razy
Wielomian czwartego stopnia
\(\displaystyle{ x^{4} - 4 x^{3} + 1 = 0\\
\left(x^{4} - 4 x^{3} \right) -\left( -1\right) =0\\
\left(x^{4} - 4 x^{3}+4x^{2} \right)-\left( 4x^{2}-1\right)=0\\
\left( x^2-2x\right)^2-\left(4x^{2}-1 \right)=0\\
\left( x^2-2x+\frac{y}{2}\right)^2-\left( \left( y+4\right)x^2-2yx+ \frac{y^2}{4}-1 \right)=0\\
\left( y^2-4\right)\left( y+4\right)-4y^2=0\\
y^3-4y-16=0\\
y=u+v\\
\left( u+v\right)^3-4\left( u+v\right)-16=0\\
u^3+3u^2v+3uv^2+v^3-4\left( u+v\right)-16\\
u^3+v^3-16+3\left( u+v\right)\left( uv- \frac{4}{3} \right)=0\\
\begin{cases} u^3+v^3-16=0 \\ 3\left( u+v\right)\left( uv- \frac{4}{3} \right)=0 \end{cases} \\
\begin{cases} u^3+v^3=16 \\ uv= \frac{4}{3} \end{cases} \\
\begin{cases} u^3+v^3=16 \\ u^3v^3= \frac{64}{27} \end{cases} \\
t^2-16t+ \frac{64}{27}=0\\
\left( t-8\right)^2-\frac{1664}{27}=0\\
\left( t-8\right)^2-\frac{4992}{81}=0\\
\left( t- \frac{72- \sqrt{4992} }{9} \right)\left( t- \frac{72+ \sqrt{4992} }{9} \right)\\
y= \frac{1}{3}\left( \sqrt[3]{288-3 \sqrt{4992} } + \sqrt[3]{288+3 \sqrt{4992}} \right)\\
\left( x^2-2x+\frac{y}{2}\right)^2-\left( \left( y+4\right)x^2-2yx+ \frac{y^2}{4}-1 \right)=0\\
\left( x^2-2x+ \frac{y}{2} \right)^2-\left( \sqrt{y+4} \right)^2\left( x- \frac{y}{ y+4 } \right)^2=0\\
\left( x^2+\left( -2- \sqrt{y+4} \right)x+ \frac{y}{2}+ \frac{y}{ \sqrt{y+4} } \right)\left( x^2+\left( -2+ \sqrt{y+4} \right)x+ \frac{y}{2}- \frac{y}{ \sqrt{y+4} } \right)=0\\}\)
\left(x^{4} - 4 x^{3} \right) -\left( -1\right) =0\\
\left(x^{4} - 4 x^{3}+4x^{2} \right)-\left( 4x^{2}-1\right)=0\\
\left( x^2-2x\right)^2-\left(4x^{2}-1 \right)=0\\
\left( x^2-2x+\frac{y}{2}\right)^2-\left( \left( y+4\right)x^2-2yx+ \frac{y^2}{4}-1 \right)=0\\
\left( y^2-4\right)\left( y+4\right)-4y^2=0\\
y^3-4y-16=0\\
y=u+v\\
\left( u+v\right)^3-4\left( u+v\right)-16=0\\
u^3+3u^2v+3uv^2+v^3-4\left( u+v\right)-16\\
u^3+v^3-16+3\left( u+v\right)\left( uv- \frac{4}{3} \right)=0\\
\begin{cases} u^3+v^3-16=0 \\ 3\left( u+v\right)\left( uv- \frac{4}{3} \right)=0 \end{cases} \\
\begin{cases} u^3+v^3=16 \\ uv= \frac{4}{3} \end{cases} \\
\begin{cases} u^3+v^3=16 \\ u^3v^3= \frac{64}{27} \end{cases} \\
t^2-16t+ \frac{64}{27}=0\\
\left( t-8\right)^2-\frac{1664}{27}=0\\
\left( t-8\right)^2-\frac{4992}{81}=0\\
\left( t- \frac{72- \sqrt{4992} }{9} \right)\left( t- \frac{72+ \sqrt{4992} }{9} \right)\\
y= \frac{1}{3}\left( \sqrt[3]{288-3 \sqrt{4992} } + \sqrt[3]{288+3 \sqrt{4992}} \right)\\
\left( x^2-2x+\frac{y}{2}\right)^2-\left( \left( y+4\right)x^2-2yx+ \frac{y^2}{4}-1 \right)=0\\
\left( x^2-2x+ \frac{y}{2} \right)^2-\left( \sqrt{y+4} \right)^2\left( x- \frac{y}{ y+4 } \right)^2=0\\
\left( x^2+\left( -2- \sqrt{y+4} \right)x+ \frac{y}{2}+ \frac{y}{ \sqrt{y+4} } \right)\left( x^2+\left( -2+ \sqrt{y+4} \right)x+ \frac{y}{2}- \frac{y}{ \sqrt{y+4} } \right)=0\\}\)