a:
\(x^4+2x^3-6x^2+2x+1=0\)
=>\(x^4-x^3+3x^3-3x^2-3x^2+3x-x+1=0\)
=>\(\left(x-1\right)\left(x^3+3x^2-3x-1\right)=0\)
=>\(\left(x-1\right)\left(x^3-x^2+4x^2-4x+x-1\right)=0\)
=>\(\left(x-1\right)^2\cdot\left(x^2+4x+1\right)=0\)
Th1: \(\left(x-1\right)^2=0\)
=>x-1=0
=>x=1
TH2: \(x^2+4x+1=0\)
=>\(x^2+4x+4-3=0\)
=>\(\left(x+2\right)^2=3\)
=>\(\left[\begin{array}{l}x+2=\sqrt3\\ x+2=-\sqrt3\end{array}\right.\Rightarrow\left[\begin{array}{l}x=\sqrt3-2\\ x=-\sqrt3-2\end{array}\right.\)
\(x^4-x^3-x+1=0\)
=>\(x^3\left(x-1\right)-\left(x-1\right)=0\)
=>\(\left(x-1\right)\left(x^3-1\right)=0\)
=>\(\left(x-1\right)^2\cdot\left(x^2+x+1\right)=0\)
=>\(\left(x-1\right)^2=0\)
=>x-1=0
=>x=1
b: \(x^5-3x^4+6x^3+6x^2-3x+1=0\)
=>\(x^5+x^4-4x^4-4x^3+10x^3+10x^2-4x^2-4x+x+1=0\)
=>(x+1)(\(\left(x^4-4x^3+10x^2-4x+1\right)=0\)
=>x+1=0
=>x=-1
Ta có: \(x^5-5x^4+4x^3+4x^2-5x+1=0\)
=>\(x^5-x^4-4x^4+4x^3+4x^2-4x-x+1=0\)
=>\(\left(x-1\right)\left(x^4-4x^3+4x-1\right)=0\)
=>\(\left(x-1\right)\left\lbrack\left(x^2-1\right)\left(x^2+1\right)-4x\left(x^2-1\right)\right\rbrack=0\)
=>\(\left(x-1\right)\left(x^2-1\right)\left(x^2-4x+1\right)=0\)
=>\(\left(x-1\right)^2\cdot\left(x+1\right)\left(x^2-4x+1\right)=0\)
TH1: \(\left(x-1\right)^2=0\)
=>x-1=0
=>x=1
TH2: x+1=0
=>x=-1
TH3: \(x^2-4x+1=0\)
=>\(x^2-4x+4=3\)
=>\(\left(x-2\right)^2=3\)
=>\(\left[\begin{array}{l}x-2=\sqrt3\\ x-2=-\sqrt3\end{array}\right.\Rightarrow\left[\begin{array}{l}x=2+\sqrt3\\ x=2-\sqrt3\end{array}\right.\)
