1: \(B=\dfrac{2x+1-x^2+2x^2-3x-1}{x\left(2x+1\right)}=\dfrac{x^2-x}{x\left(2x+1\right)}=\dfrac{x-1}{2x+1}\)

2: \(C=A:B\)

\(=\dfrac{x-1}{x^2}:\dfrac{x-1}{2x+1}=\dfrac{2x+1}{x^2}\)

\(C+1=\dfrac{2x+1+x^2}{x^2}=\dfrac{\left(x+1\right)^2}{x^2}>=0\)

=>C>=-1

Bài 1:

a. $2x^3+3x^2-2x=2x(x^2+3x-2)=2x[(x^2-2x)+(x-2)]$

$=2x[x(x-2)+(x-2)]=2x(x-2)(x+1)$

b.

$(x+1)(x+2)(x+3)(x+4)-24$

$=[(x+1)(x+4)][(x+2)(x+3)]-24$

$=(x^2+5x+4)(x^2+5x+6)-24$

$=a(a+2)-24$ (đặt $x^2+5x+4=a$)

$=a^2+2a-24=(a^2-4a)+(6a-24)$

$=a(a-4)+6(a-4)=(a-4)(a+6)=(x^2+5x)(x^2+5x+10)$

$=x(x+5)(x^2+5x+10)$

Bài 2:

a. ĐKXĐ: $x\neq 3; 4$

\(A=\frac{2x+1-(x+3)(x-3)+(2x-1)(x-4)}{(x-3)(x-4)}\\ =\frac{2x+1-(x^2-9)+(2x^2-9x+4)}{(x-3)(x-4)}\\ =\frac{x^2-7x+14}{(x-3)(x-4)}\)

b. $x^2+20=9x$

$\Leftrightarrow x^2-9x+20=0$

$\Leftrightarrow (x-4)(x-5)=0$

$\Rightarrow x=5$ (do $x\neq 4$)

Khi đó: $A=\frac{5^2-7.5+14}{(5-4)(5-3)}=2$

ĐKXĐ : \(x\ne\left\{1;0\right\}\)

a) \(P=\left(\dfrac{\left(x-1\right)^2}{3x+\left(x-1\right)^2}-\dfrac{1-2x^2+4x}{x^3-1}+\dfrac{1}{x-1}\right):\dfrac{2x}{x^3+x}\)

\(P=\left(\dfrac{\left(x-1\right)^2}{x^2+x+1}-\dfrac{1-2x^2+4x}{\left(x-1\right)\left(x^2+x+1\right)}+\dfrac{1}{x-1}\right)\cdot\dfrac{x\left(x^2+1\right)}{2x}\)

\(P=\left(\dfrac{\left(x-1\right)\left(x-1\right)^2}{\left(x-1\right)\left(x^2+x+1\right)}-\dfrac{1-2x^2+4x}{\left(x-1\right)\left(x^2+x+1\right)}+\dfrac{x^2+x+1}{\left(x-1\right)\left(x^2+x+1\right)}\right)\cdot\dfrac{x^2+1}{2}\)

\(P=\left(\dfrac{\left(x-1\right)^3-1+2x^2-4x+x^2+x+1}{\left(x-1\right)\left(x^2+x+1\right)}\right)\cdot\dfrac{x^2+1}{2}\)

\(P=\left(\dfrac{x^3-3x^2+3x-1-1+2x^2-4x+x^2+x+1}{\left(x-1\right)\left(x^2+x+1\right)}\right)\cdot\dfrac{x^2+1}{2}\)

\(P=\left(\dfrac{x^3-1}{x^3-1}\right)\cdot\dfrac{x^2+1}{2}\)

\(P=1\cdot\dfrac{x^2+1}{2}\)

\(P=\dfrac{x^2+1}{2}\)

b) Vì \(x^2\ge0\forall x\)

\(\Rightarrow P\ge\dfrac{1}{2}\)

Dấu "=" xảy ra \(\Leftrightarrow x=0\)

Mà ĐKXĐ \(x\ne0\)

=> ... đến đây ko biết làm :v

AI BIẾT LÀM HỘ ĐI

Cái này mk chưa học nên cx chưa rõ cách làm chính xác mong bạn thông cảm :)

\(P=\left(\dfrac{\left(x-1\right)^2}{x^2+x+1}+\dfrac{2x^2-4x-1}{\left(x-1\right)\left(x^2+x+1\right)}+\dfrac{1}{x-1}\right)\cdot\dfrac{x\left(x^2+1\right)}{2x}\)

\(=\dfrac{x^3-3x^2+3x-1+2x^2-4x-1+x^2+x+1}{\left(x-1\right)\left(x^2+x+1\right)}\cdot\dfrac{x^2+1}{2}\)

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

a, \(\left(\dfrac{x}{x^2-49}-\dfrac{x-7}{x^2+7x}\right):\dfrac{2x-7}{x^2+7x}+\dfrac{x}{7-x}\)

\(=\left[\dfrac{x^2}{x\left(x-7\right)\left(x+7\right)}-\dfrac{\left(x-7\right)^2}{x\left(x+7\right)\left(x-7\right)}\right].\dfrac{x\left(x+7\right)}{2x-7}+\dfrac{x}{7-x}\)

\(=\left[\dfrac{x^2-\left(x-7\right)^2}{x\left(x-7\right)\left(x+7\right)}\right].\dfrac{x\left(x+7\right)}{2x-7}+\dfrac{x}{7-x}\)

\(=\left[\dfrac{\left(x-x+7\right)\left(x+x-7\right)}{x\left(x-7\right)\left(x+7\right)}\right].\dfrac{x\left(x+7\right)}{2x-7}+\dfrac{x}{7-x}\)

\(=\dfrac{7\left(2x-7\right)}{x\left(x-7\right)\left(x+7\right)}.\dfrac{x\left(x+7\right)}{2x-7}+\dfrac{x}{7-x}\)

\(=\dfrac{7}{x-7}+\dfrac{x}{7-x}\)

\(=\dfrac{7\left(7-x\right)+x\left(x-7\right)}{\left(x-7\right)\left(7-x\right)}=\dfrac{49-7x+x^2-7x}{7x-x^2-49+7x}\)

\(=\dfrac{49-14x+x^2}{14x-x^2-49}=\dfrac{-\left(14-x^2-49\right)}{14x-x^2-49}=-1\)

\(\Rightarrow\)Giá trị biểu thức trên không phụ thuộc vào biến

\(\Rightarrowđpcm\)

b, tương tự

Bạn chỉ cần tính ra thôi .

+ Nếu kết quả rút gọn có x thì kết luận giá trị của biểu thức phụ thuộc vào giá trị của biến .

+ Nếu kết quả rút gọn không có x thì kết luận giá trị của biểu thức không phụ thuộc vào giá trị của biến .

Chúc bạn làm bài tốt nha .

a: \(A=\left(\dfrac{2x^2+2}{x^3-1}+\dfrac{x^2-x+1}{x^4+x^2+1}-\dfrac{x^2+3}{x^3-x^2+3x-3}\right):\dfrac{1}{x-1}\)

\(=\left(\dfrac{2x^2+2}{\left(x-1\right)\left(x^2+x+1\right)}+\dfrac{x^2-x+1}{x^4+2x^2+1-x^2}-\dfrac{x^2+3}{x^2\left(x-1\right)+3\left(x-1\right)}\right)\cdot\dfrac{x-1}{1}\)

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

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

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

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

\(=\dfrac{x^2+1}{x^2+x+1}\)

b: Để A là số nguyên thì \(x^2+1⋮x^2+x+1\)

=>\(x^2+x+1-x⋮x^2+x+1\)

=>\(x⋮x^2+x+1\)

=>\(x^2+x⋮x^2+x+1\)

=>\(x^2+x+1-1⋮x^2+x+1\)

=>\(-1⋮x^2+x+1\)

=>\(x^2+x+1\in\left\{1;-1\right\}\)

=>\(x^2+x+1=1\)

=>x²+x=0

=>x(x+1)=0

=>\(x\in\left\{0;-1\right\}\)