=>x(x^2-9/16)=0

=>x(x-3/4)(x+3/4)=0

=>x=0; x=3/4; x=-3/4

Đặt 2005 = x +1 . Ta có :

x⁶ - (x + 1 )x⁵ + ( x + 1 )x⁴ - (x + 1 )x³ + ( x + 1 )x² - (x + 1)x + (x + 1)

= x⁶ - x⁶ - x⁵ + x⁵ + x⁴ - x⁴ - x³ + x³ + x² - x² -x + x + 1

= 1

Bằng 0 chứ nhỉ em ?

(x-5) . (2x-4)= 0

\(\left[{}\begin{matrix}x-5=0\\2x-4=0\end{matrix}\right.< =>\left[{}\begin{matrix}x=5\\2x=4\end{matrix}\right.< =>\left[{}\begin{matrix}x=5\\x=2\end{matrix}\right.\)

B= (1-1/2). ( 1-1/3).(1-1/4).(1-1/5)....(1-1/2004)

B= 1/2. 2/3 . 3/4. 4/5....2003/2004

B= 1/2004

\(B=\left(1-\frac{1}{2}\right)\left(1-\frac{1}{3}\right)\left(1-\frac{1}{4}\right)...\left(1-\frac{1}{2003}\right)\left(1-\frac{1}{2004}\right)\)

\(B=\frac{1}{2}\cdot\frac{2}{3}\cdot\frac{3}{4}\cdot...\cdot\frac{2002}{2003}\cdot\frac{2003}{2004}\)

\(B=\frac{1}{2004}\)

\(B=\left(1-\frac{2}{2}\right).\left(1-\frac{1}{3}\right).\left(1-\frac{1}{4}\right).\left(1-\frac{1}{5}\right).....\left(1-\frac{1}{2003}\right).\left(1-\frac{1}{2004}\right)\)

\(=\frac{1}{2}.\frac{2}{3}.\frac{3}{4}....\frac{2002}{2003}.\frac{2003}{2004}\)

\(=\frac{1}{2004}\)

a/ \(x^2\left(x-5\right)+5-x=0\)

\(\Leftrightarrow x^2\left(x-5\right)-\left(x-5\right)=0\)

\(\Leftrightarrow\left(x-5\right)\left(x^2-1\right)=0\)

\(\Leftrightarrow\left(x-1\right)\left(x+1\right)\left(x-5\right)=0\)

\(\Leftrightarrow\left[{}\begin{matrix}x-1=0\\x+1=0\\x-5=0\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}x=1\\x=-1\\x=5\end{matrix}\right.\)

Vậy...

b/ \(3x^4-9x^3=-9x^2+27x\)

\(\Leftrightarrow3x^4-9x^3+9x^2-27x=0\)

\(\Leftrightarrow3x^3\left(x-3\right)+9x\left(x-3\right)=0\)

\(\Leftrightarrow\left(x-3\right)\left(3x^3+9x\right)=0\)

\(\Leftrightarrow3x\left(x-3\right)\left(x^2+3\right)=0\)

Vì \(x^2+3>0\forall x\)

\(\Leftrightarrow3x\left(x-3\right)=0\)

\(\Leftrightarrow\left[{}\begin{matrix}x=0\\x-3=0\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}x=0\\x=3\end{matrix}\right.\)

Vậy..

c/ \(x^2\left(x+8\right)+x^2=-8x\)

\(\Leftrightarrow x^2\left(x+8\right)+x^2+8x=0\)

\(\Leftrightarrow x^2\left(x+8\right)+x\left(x+8\right)=0\)

\(\Leftrightarrow x\left(x+8\right)\left(x+1\right)=0\)

\(\Leftrightarrow\left[{}\begin{matrix}x=0\\x+8=0\\x+1=0\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}x=0\\x=-8\\x=-1\end{matrix}\right.\)

Vậy...

d/ \(\left(x+3\right)\left(x^2-3x+5\right)=x^2+3x\)

\(\Leftrightarrow\left(x+3\right)\left(x^2-3x+5\right)-x\left(x+3\right)=0\)

\(\Leftrightarrow\left(x+3\right)\left(x^2-4x+5\right)=0\)

\(\Leftrightarrow\left(x+3\right)\left[\left(x-2\right)^2+1\right]=0\)

Vì \(\left(x-2\right)^2+1>0\forall x\)

\(\Leftrightarrow x+3=0\Leftrightarrow x=-3\)

Vậy..

\(a,x^2\left(x-5\right)+5-x=0\\ \Leftrightarrow x^2\left(x-5\right)-\left(x-5\right)=0\\ \Leftrightarrow\left(x^2-1\right)\left(x-5\right)=0\\ \Leftrightarrow\left[{}\begin{matrix}x=1\\x=-1\\x=5\end{matrix}\right.\)

\(b,3x^4-9x^3=-9x^2+27x\\ \Leftrightarrow3x^4-9x^3+9x^2-27x=0\\ \Leftrightarrow3x^3\left(x-3\right)+9x\left(x-3\right)=0\\ \Leftrightarrow3x\left(x-3\right)\left(x^2+2\right)=0\\ \Leftrightarrow\left[{}\begin{matrix}x=0\\x=3\end{matrix}\right.\)

do \(x^2+2>0\)

\(c,x^2\left(x+8\right)+x^2=-8x\\ \Leftrightarrow x^2\left(x+8\right)+x^2+8x=0\\ \Leftrightarrow x^2\left(x+8\right)+x\left(x+8\right)=0\\ \Leftrightarrow x\left(x+1\right)\left(x+8\right)=0\\ \Leftrightarrow\left[{}\begin{matrix}x=0\\x=-1\\x=-8\end{matrix}\right.\)

\(d,\left(x+3\right)\left(x^2-3x+5\right)=x^2+3x\\ \Leftrightarrow\left(x+3\right)\left(x^2-3x+5\right)-x^2-3x=0\\ \Leftrightarrow\left(x+3\right)\left(x^2-3x+5\right)-x\left(x+3\right)=0\\ \Leftrightarrow\left(x+3\right)\left(x^2-3x+5-x\right)=0\\ \Leftrightarrow\left(x+3\right)\left(x^2-4x+5\right)=0\\ \Leftrightarrow\left[{}\begin{matrix}x=-3\\x^2-4x+5=0\end{matrix}\right.\)

\(\Delta=b^2-4ac=\left(-4\right)^2-4\cdot1\cdot5=16-20=-4< 0\)\(\rightarrow\)\(x^2-4x+5\) vô nghiệm

Vậy \(x=-3\)

Bài 4:

a) Ta có: \(x^9-x^7-x^6-x^5+x^4+x^3+x^2-1\)

\(=\left(x^9-x^7\right)-\left(x^6-x^4\right)-\left(x^5-x^3\right)+\left(x^2-1\right)\)

\(=x^7\left(x^2-1\right)-x^4\left(x^2-1\right)-x^3\left(x^2-1\right)+\left(x^2-1\right)\)

\(=\left(x^2-1\right)\left(x^7-x^4-x^3+1\right)\)

\(=\left(x^2-1\right)\cdot\left[x^4\left(x^3-1\right)-\left(x^3-1\right)\right]\)

\(=\left(x^2-1\right)\cdot\left(x^3-1\right)\cdot\left(x^4-1\right)\)

\(=\left(x-1\right)\left(x+1\right)\cdot\left(x-1\right)\left(x^2+x+1\right)\cdot\left(x-1\right)\left(x+1\right)\left(x^2+1\right)\)

\(=\left(x-1\right)^3\cdot\left(x+1\right)^2\cdot\left(x^2+1\right)\cdot\left(x^2+x+1\right)\)

a, Ta có : \(x^5-x^4-x^3-x^2-x-2\)

\(=x^5-2x^4+x^4-2x^3+x^3-2x^2+x^2-2x+x-2\)

\(=x^4\left(x-2\right)+x^3\left(x-2\right)+x^2\left(x-2\right)+x\left(x-2\right)+\left(x-2\right)\)

\(=\left(x-2\right)\left(x^4+x^3+x^2+x+1\right)\)