Znajdź pierwiastki wielomianu
\(\displaystyle{ x ^{4} + 21x ^{3} + 60x ^{2} + 59x - 42 = 0}\)
Znajdź pierwiastki wielomianu
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Vax
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Znajdź pierwiastki wielomianu
Dobrze przepisałaś/eś przykład ? Dany wielomian nie ma pierwiastków wymiernych, jeżeli przykład jest ok pozostaje metoda Ferrariego.
Pozdrawiam.
Pozdrawiam.
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Mariusz M
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Znajdź pierwiastki wielomianu
\(\displaystyle{ x ^{4} + 21x ^{3} + 60x ^{2} + 59x - 42 = 0\\
x^{4}+21x^{3}=-60x^{2}-59x+42\\
x^{4}+21x^{3}+ \frac{441}{4}x^{2}= \frac{201}{4}x^{2}-59x+42\\
\left( x^{2}+ \frac{21}{2}x \right)^{2}= \frac{201}{4}x^{2}-59x+42\\
\left( x^{2}+ \frac{21}{2}x+ \frac{y}{2} \right)^{2}=\left( y+ \frac{201}{4} \right)x^{2}+\left( \frac{21}{2}y-59 \right)x+\frac{y^{2}}{4}+42\\
\left( y^{2}+168\right)\left( y+ \frac{201}{4} \right)-\left( \frac{21}{2}y-59 \right)^{2}=0\\
y^{3}+ \frac{201}{4}y^{2}+168y+8442- \frac{441}{4}y^{2} +1239y-3481=0\\
y^{3}-60y^{2}+1407y+4961=0\\
y=w+20\\
\left( w+20\right)^{3}-60\left( w+20\right)^{2}+1407\left( w+20\right)+4961=0\\
w^{3}+60w^{2}+1200w+8000-60w^{2}-2400w-24000+1407w+28140+4961=0\\
w^{3}+207w+17101=0\\
w=u+v\\
\left( u+v\right)^3+207\left( u+v\right)+17101=0\\
u^3+v^3+3u^2v+3uv^2+207\left( u+v\right)+17101=0\\
\begin{cases} u^3+v^3+17101=0 \\ 3\left( u+v\right)\left( uv+69\right)=0 \end{cases} \\
\begin{cases} u^3+v^3+17101=0 \\ \left( uv+69\right)=0 \end{cases} \\
\begin{cases} u^3+v^3=-17101 \\ uv=-69 \end{cases} \\
\begin{cases} u^3+v^3=-17101 \\ u^3v^3=-328509 \end{cases} \\
t^2+17101t-328509=0\\
\Delta=17101^2+4 \cdot 328509=292444201+1314036=293758237\\
t_{1}= \frac{-17101- \sqrt{293758237} }{2} \\
t_{2}= \frac{-17101+ \sqrt{293758237} }{2} \\
w= \sqrt[3]{\frac{-17101- \sqrt{293758237} }{2}}+ \sqrt[3]{\frac{-17101+ \sqrt{293758237} }{2}}\\
y= \sqrt[3]{\frac{-17101- \sqrt{293758237} }{2}}+ \sqrt[3]{\frac{-17101+ \sqrt{293758237} }{2}}+20\\}\)
\(\displaystyle{ \left( x^{2}+ \frac{21}{2}x+ \frac{y}{2} \right)^{2}=\left( \frac{ \sqrt{4y+201} }{2}x- \frac{ \sqrt{y^{2}+168} }{2} \right)^{2}\\
\left( x^{2}+ \frac{21}{2}x+ \frac{y}{2} \right)^{2}-\left( \frac{ \sqrt{4y+201} }{2}x- \frac{ \sqrt{y^{2}+168} }{2} \right)^{2}=0\\
\left( x^{2}+ \frac{21- \sqrt{4y+201} }{2}x+ \frac{y+ \sqrt{y^{2}+168} }{2} \right)\left( x^{2}+ \frac{21- \sqrt{4y+201} }{2}x+ \frac{y+ \sqrt{y^{2}+168} }{2} \right)=0\\
\left( 2x^{2}+\left( 21- \sqrt{4y+201} \right)x+\left( y+ \sqrt{y^2+168} \right) \right)\left( 2x^{2}+\left( 21+ \sqrt{4y+201} \right)x+\left( y- \sqrt{y^2+168} \right) \right) =0\\
\Delta_{1}=441-42\sqrt{4y+201}+4y+201-8\left( y+ \sqrt{y^2+168} \right)\\
= 642-4y-42\sqrt{4y+201}-8 \sqrt{y^2+168}\\
x_{1}= \frac{-21+ \sqrt{4y+201}- \sqrt{642-4y-42\sqrt{4y+201}-8 \sqrt{y^2+168}} }{4}\\
x_{2}= \frac{-21+ \sqrt{4y+201}+ \sqrt{642-4y-42\sqrt{4y+201}-8 \sqrt{y^2+168}} }{4}\\
\Delta_{2}=441+42 \sqrt{4y+201}+4y+201-8\left( y- \sqrt{y^2+168} \right)\\
=642-4y+42 \sqrt{4y+201}+8 \sqrt{y^2+168}\\
x_{3}= \frac{-21- \sqrt{4y+201}- \sqrt{642-4y+42 \sqrt{4y+201}+8 \sqrt{y^2+168}} }{4}\\
x_{4}= \frac{-21- \sqrt{4y+201}+ \sqrt{642-4y+42 \sqrt{4y+201}+8 \sqrt{y^2+168}} }{4}\\}\)
\(\displaystyle{ y= \sqrt[3]{\frac{-17101- \sqrt{293758237} }{2}}+ \sqrt[3]{\frac{-17101+ \sqrt{293758237} }{2}}+20\\
x_{1}= \frac{-21+ \sqrt{4y+201}- \sqrt{642-4y-42\sqrt{4y+201}-8 \sqrt{y^2+168}} }{4}\\
x_{2}= \frac{-21+ \sqrt{4y+201}+ \sqrt{642-4y-42\sqrt{4y+201}-8 \sqrt{y^2+168}} }{4}\\
x_{3}= \frac{-21- \sqrt{4y+201}- \sqrt{642-4y+42 \sqrt{4y+201}+8 \sqrt{y^2+168}} }{4}\\
x_{4}= \frac{-21- \sqrt{4y+201}+ \sqrt{642-4y+42 \sqrt{4y+201}+8 \sqrt{y^2+168}} }{4}\\}\)
x^{4}+21x^{3}=-60x^{2}-59x+42\\
x^{4}+21x^{3}+ \frac{441}{4}x^{2}= \frac{201}{4}x^{2}-59x+42\\
\left( x^{2}+ \frac{21}{2}x \right)^{2}= \frac{201}{4}x^{2}-59x+42\\
\left( x^{2}+ \frac{21}{2}x+ \frac{y}{2} \right)^{2}=\left( y+ \frac{201}{4} \right)x^{2}+\left( \frac{21}{2}y-59 \right)x+\frac{y^{2}}{4}+42\\
\left( y^{2}+168\right)\left( y+ \frac{201}{4} \right)-\left( \frac{21}{2}y-59 \right)^{2}=0\\
y^{3}+ \frac{201}{4}y^{2}+168y+8442- \frac{441}{4}y^{2} +1239y-3481=0\\
y^{3}-60y^{2}+1407y+4961=0\\
y=w+20\\
\left( w+20\right)^{3}-60\left( w+20\right)^{2}+1407\left( w+20\right)+4961=0\\
w^{3}+60w^{2}+1200w+8000-60w^{2}-2400w-24000+1407w+28140+4961=0\\
w^{3}+207w+17101=0\\
w=u+v\\
\left( u+v\right)^3+207\left( u+v\right)+17101=0\\
u^3+v^3+3u^2v+3uv^2+207\left( u+v\right)+17101=0\\
\begin{cases} u^3+v^3+17101=0 \\ 3\left( u+v\right)\left( uv+69\right)=0 \end{cases} \\
\begin{cases} u^3+v^3+17101=0 \\ \left( uv+69\right)=0 \end{cases} \\
\begin{cases} u^3+v^3=-17101 \\ uv=-69 \end{cases} \\
\begin{cases} u^3+v^3=-17101 \\ u^3v^3=-328509 \end{cases} \\
t^2+17101t-328509=0\\
\Delta=17101^2+4 \cdot 328509=292444201+1314036=293758237\\
t_{1}= \frac{-17101- \sqrt{293758237} }{2} \\
t_{2}= \frac{-17101+ \sqrt{293758237} }{2} \\
w= \sqrt[3]{\frac{-17101- \sqrt{293758237} }{2}}+ \sqrt[3]{\frac{-17101+ \sqrt{293758237} }{2}}\\
y= \sqrt[3]{\frac{-17101- \sqrt{293758237} }{2}}+ \sqrt[3]{\frac{-17101+ \sqrt{293758237} }{2}}+20\\}\)
\(\displaystyle{ \left( x^{2}+ \frac{21}{2}x+ \frac{y}{2} \right)^{2}=\left( \frac{ \sqrt{4y+201} }{2}x- \frac{ \sqrt{y^{2}+168} }{2} \right)^{2}\\
\left( x^{2}+ \frac{21}{2}x+ \frac{y}{2} \right)^{2}-\left( \frac{ \sqrt{4y+201} }{2}x- \frac{ \sqrt{y^{2}+168} }{2} \right)^{2}=0\\
\left( x^{2}+ \frac{21- \sqrt{4y+201} }{2}x+ \frac{y+ \sqrt{y^{2}+168} }{2} \right)\left( x^{2}+ \frac{21- \sqrt{4y+201} }{2}x+ \frac{y+ \sqrt{y^{2}+168} }{2} \right)=0\\
\left( 2x^{2}+\left( 21- \sqrt{4y+201} \right)x+\left( y+ \sqrt{y^2+168} \right) \right)\left( 2x^{2}+\left( 21+ \sqrt{4y+201} \right)x+\left( y- \sqrt{y^2+168} \right) \right) =0\\
\Delta_{1}=441-42\sqrt{4y+201}+4y+201-8\left( y+ \sqrt{y^2+168} \right)\\
= 642-4y-42\sqrt{4y+201}-8 \sqrt{y^2+168}\\
x_{1}= \frac{-21+ \sqrt{4y+201}- \sqrt{642-4y-42\sqrt{4y+201}-8 \sqrt{y^2+168}} }{4}\\
x_{2}= \frac{-21+ \sqrt{4y+201}+ \sqrt{642-4y-42\sqrt{4y+201}-8 \sqrt{y^2+168}} }{4}\\
\Delta_{2}=441+42 \sqrt{4y+201}+4y+201-8\left( y- \sqrt{y^2+168} \right)\\
=642-4y+42 \sqrt{4y+201}+8 \sqrt{y^2+168}\\
x_{3}= \frac{-21- \sqrt{4y+201}- \sqrt{642-4y+42 \sqrt{4y+201}+8 \sqrt{y^2+168}} }{4}\\
x_{4}= \frac{-21- \sqrt{4y+201}+ \sqrt{642-4y+42 \sqrt{4y+201}+8 \sqrt{y^2+168}} }{4}\\}\)
\(\displaystyle{ y= \sqrt[3]{\frac{-17101- \sqrt{293758237} }{2}}+ \sqrt[3]{\frac{-17101+ \sqrt{293758237} }{2}}+20\\
x_{1}= \frac{-21+ \sqrt{4y+201}- \sqrt{642-4y-42\sqrt{4y+201}-8 \sqrt{y^2+168}} }{4}\\
x_{2}= \frac{-21+ \sqrt{4y+201}+ \sqrt{642-4y-42\sqrt{4y+201}-8 \sqrt{y^2+168}} }{4}\\
x_{3}= \frac{-21- \sqrt{4y+201}- \sqrt{642-4y+42 \sqrt{4y+201}+8 \sqrt{y^2+168}} }{4}\\
x_{4}= \frac{-21- \sqrt{4y+201}+ \sqrt{642-4y+42 \sqrt{4y+201}+8 \sqrt{y^2+168}} }{4}\\}\)