Bài 1:
a: \(x^3-9x^2+6x+16\)
\(=x^3-8x^2-x^2+8x-2x+16\)
\(=x^2\left(x-8\right)-x\left(x-8\right)-2\left(x-8\right)\)
\(=\left(x-8\right)\left(x^2-x-2\right)\)
\(=\left(x-8\right)\left(x^2-2x+x-2\right)\)
\(=\left(x-8\right)\left[x\left(x-2\right)+\left(x-2\right)\right]\)
\(=\left(x-8\right)\left(x-2\right)\left(x+1\right)\)
b: \(x^3-x^2-x-2\)
\(=x^3-2x^2+x^2-2x+x-2\)
\(=x^2\left(x-2\right)+x\left(x-2\right)+\left(x-2\right)\)
\(=\left(x-2\right)\left(x^2+x+1\right)\)
c: \(x^3-4x^2-3x+18\)
\(=x^3+2x^2-6x^2-12x+9x+18\)
\(=x^2\left(x+2\right)-6x\left(x+2\right)+9\left(x+2\right)\)
\(=\left(x+2\right)\left(x^2-6x+9\right)\)
\(=\left(x+2\right)\left(x-3\right)^2\)
Bài 2:
a: \(x^3+5x^2+3x-9=0\)
=>\(x^3-x^2+6x^2-6x+9x-9=0\)
=>\(x^2\left(x-1\right)+6x\left(x-1\right)+9\left(x-1\right)=0\)
=>\(\left(x-1\right)\left(x^2+6x+9\right)=0\)
=>\(\left(x-1\right)\left(x+3\right)^2=0\)
=>\(\left[{}\begin{matrix}x-1=0\\\left(x+3\right)^2=0\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}x-1=0\\x+3=0\end{matrix}\right.\)
=>\(\left[{}\begin{matrix}x=1\\x=-3\end{matrix}\right.\)
b: \(x^3-4x^2+x+6=0\)
=>\(x^3+x^2-5x^2-5x+6x+6=0\)
=>\(x^2\left(x+1\right)-5x\left(x+1\right)+6\left(x+1\right)=0\)
=>\(\left(x+1\right)\left(x^2-5x+6\right)=0\)
=>\(\left(x+1\right)\left(x-2\right)\left(x-3\right)=0\)
=>\(\left[{}\begin{matrix}x+1=0\\x-2=0\\x-3=0\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}x=-1\\x=2\\x=3\end{matrix}\right.\)
c: \(x^3-9x^2+26x-24=0\)
=>\(x^3-2x^2-7x^2+14x+12x-24=0\)
=>\(x^2\left(x-2\right)-7x\left(x-2\right)+12\left(x-2\right)=0\)
=>\(\left(x-2\right)\left(x^2-7x+12\right)=0\)
=>(x-2)(x-3)(x-4)=0
=>\(\left[{}\begin{matrix}x-2=0\\x-3=0\\x-4=0\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}x=2\\x=3\\x=4\end{matrix}\right.\)
Bài 3:
a: \(\left(x^2+x+1\right)\left(x^2+x+2\right)-12\)
\(=\left(x^2+x\right)^2+2\left(x^2+x\right)+\left(x^2+x\right)+2-12\)
\(=\left(x^2+x\right)^2+3\left(x^2+x\right)-10\)
\(=\left(x^2+x+5\right)\left(x^2+x-2\right)\)
\(=\left(x^2+x+5\right)\left(x^2+2x-x-2\right)\)
\(=\left(x^2+x+5\right)\left(x+2\right)\left(x-1\right)\)
b: \(\left(x+2\right)\left(x+3\right)\left(x+4\right)\left(x+5\right)-24\)
\(=\left[\left(x+2\right)\left(x+5\right)\right]\left[\left(x+3\right)\left(x+4\right)\right]-24\)
\(=\left(x^2+7x+10\right)\left(x^2+7x+12\right)-24\)
\(=\left(x^2+7x\right)^2+12\left(x^2+7x\right)+10\left(x^2+7x\right)+120-24\)
\(=\left(x^2+7x\right)^2+22\left(x^2+7x\right)+96\)
\(=\left(x^2+7x\right)^2+6\left(x^2+7x\right)+16\left(x^2+7x\right)+96\)
\(=\left(x^2+7x+6\right)\left(x^2+7x+16\right)\)
\(=\left(x+1\right)\left(x+6\right)\left(x^2+7x+16\right)\)
