giúp mình với mn ơi $\begin{array}la)~Rút~gọn~biểu~thức~P=(\frac{2x\sqrt x+x-\sqrt x}{2\sqrt x-1}).\frac{x+\sqrt x}{2x+\sqrt x-1}+\frac{\sqrt x}{2\sqrt x-1}~oớ~(2\geq0,x+1,x+\frac14)\b)~cho~l(x)~là~triếthắc~bộn~thoả~mồn~P(-2)=0~và~P(x)-P(x-2)=x(x+2)~(x+1)\end{array}$

Question

Accepted Answer

a,
$\displaystyle \begin{array}{{>{\displaystyle}l}}
P=\left(\frac{2x\sqrt{x} +x-\sqrt{x}}{x\sqrt{x} -1} -\frac{x+\sqrt{x}}{x-1} ight) .\frac{x-1}{2x+\sqrt{x} -1} +\frac{\sqrt{x}}{2\sqrt{x} -1}\
P=\left(\frac{\sqrt{x}\left( 2x+\sqrt{x} -1 ight)}{\left(\sqrt{x} -1 ight)\left( x+\sqrt{x} +1 ight)} -\frac{\sqrt{x}\left(\sqrt{x} +1 ight)}{\left(\sqrt{x} -1 ight)\left(\sqrt{x} +1 ight)} ight) .\frac{\left(\sqrt{x} -1 ight)\left(\sqrt{x} +1 ight)}{2x+\sqrt{x} -1} +\frac{\sqrt{x}}{2\sqrt{x-1}}\
P=\frac{\sqrt{x} .\left(\sqrt{x} +1 ight)}{x+\sqrt{x} +1} -\frac{x+\sqrt{x}}{2x+\sqrt{x} -1} +\frac{\sqrt{x}}{2\sqrt{x} -1}\
P=\frac{x+\sqrt{x}}{x+\sqrt{x} +1} -\frac{x+\sqrt{x}}{\left( 2\sqrt{x} -1 ight)\left(\sqrt{x} +1 ight)} +\frac{x+\sqrt{x}}{\left( 2\sqrt{x} -1 ight)\left(\sqrt{x} +1 ight)}\
P=\frac{x+\sqrt{x}}{x+\sqrt{x} +1}
\end{array}$
b,
Ta có:
$\displaystyle \begin{array}{{>{\displaystyle}l}}
P( x) -P( x-2) =x( x+2)( 3x+1)\
\Rightarrow P( 0) -P( 0-2) =0.( 0+2)( 3.0+1) =0\
\Rightarrow P( 0) -P( -2) =0
\end{array}$
mà $\displaystyle P( -2) =0$
$\displaystyle \begin{array}{{>{\displaystyle}l}}
\Rightarrow P( 0) =P( -2) =0\
P( -2) -P( -2-2) =-2( -2+2)( -2.3+1) =0\
\Rightarrow P( -2) -P( -4) =0\
\Leftrightarrow P( -4) =0
\end{array}$
Vậy P(x) = 0 có 3 nghiệm x=0, x=-2, x=-4
$\displaystyle \Rightarrow P( x) =x( x+2)( x+4)( ax+b)$
$\displaystyle \begin{array}{{>{\displaystyle}l}}
P( 2) -P( 0) =2( 2+2)( 3.2+1) =56\
\Rightarrow P( 2) =56\
\Rightarrow 2a+b=\frac{7}{6}\
P( 4) -P( 2) =312\
\Rightarrow P( 4) =368\
\Rightarrow 3a+b=\frac{23}{12}\
\Rightarrow a=\frac{3}{4} ,\ b=\frac{-1}{3}\
\Rightarrow P( x) =x( x+2)( x+4)\left(\frac{3}{4} x-\frac{1}{3} ight)
\end{array}$