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Gọi phân thức cần tìm là \(A\)
Ta có:
\(\dfrac{1}{x}.\dfrac{x}{x+1}.\dfrac{x+1}{x+2}.\dfrac{x+2}{x+3}.\dfrac{x+3}{x+4}.\dfrac{x+4}{x+5}.\dfrac{x+5}{x+6}.\dfrac{x+6}{x+7}.\dfrac{x+7}{x+8}.\dfrac{x+8}{x+9}.\dfrac{x+9}{x+10}\)
\(=\dfrac{x\left(x+1\right)\left(x+2\right)\left(x+3\right)\left(x+4\right)\left(x+5\right)\left(x+6\right)\left(x+7\right)\left(x+8\right)\left(x+9\right)}{\left(x+1\right)\left(x+2\right)\left(x+3\right)\left(x+4\right)\left(x+5\right)\left(x+6\right)\left(x+7\right)\left(x+8\right)\left(x+9\right)\left(x+10\right)}\)\(=\dfrac{x}{x+10}\)
Suy ra:
\(\dfrac{1}{x}.\dfrac{x}{x+1}.\dfrac{x+1}{x+2}.\dfrac{x+2}{x+3}.\dfrac{x+3}{x+4}.\dfrac{x+4}{x+5}.\dfrac{x+5}{x+6}.\dfrac{x+6}{x+7}.\dfrac{x+7}{x+8}.\dfrac{x+8}{x+9}.\dfrac{x+9}{x+10}.A=1\)
\(\Leftrightarrow\dfrac{x}{x+10}.A=1\)
\(\Leftrightarrow A=\dfrac{x+10}{x}\)
Vậy phân thức cần điền vào chỗ trống là \(\dfrac{x+10}{x}\)
\(\dfrac{x}{x+1}:\dfrac{x+2}{x+3}:\dfrac{x+3}{x+4}:\dfrac{x+4}{x+5}:\dfrac{x+5}{x+6}=\dfrac{x}{x+6}\)
a.\(\dfrac{1}{x}\)-\(\dfrac{1}{x+1}\)=\(\dfrac{x+1}{x\left(x+1\right)}\)-\(\dfrac{x}{x\left(x+1\right)}\)=\(\dfrac{x+1-x}{x\left(x+1\right)}\)=\(\dfrac{1}{x\left(x+1\right)}\)
b. Ta có:
\(\dfrac{1}{x\left(x+1\right)}\)= \(\dfrac{\left(x+1\right)-x}{x\left(x+1\right)}\)=\(\dfrac{x+1}{x\left(x+1\right)}\)-\(\dfrac{x}{x\left(x+1\right)}\)=\(\dfrac{1}{x}\)-\(\dfrac{1}{x+1}\)
Ta lại có:
\(\dfrac{1}{\left(x+1\right)\left(x+2\right)}\)=\(\dfrac{1}{x+1}\)-\(\dfrac{1}{x+2}\);
\(\dfrac{1}{\left(x+2\right)\left(x+3\right)}\)=\(\dfrac{1}{x+2}\)-\(\dfrac{1}{x+3}\);
\(\dfrac{1}{\left(x+3\right)\left(x+4\right)}\)=\(\dfrac{1}{x+3}\)-\(\dfrac{1}{x+4}\);
\(\dfrac{1}{\left(x+4\right)\left(x+5\right)}\)=\(\dfrac{1}{x+4}\)-\(\dfrac{1}{x+5}\);
Do đó:
\(\dfrac{1}{x\left(x+1\right)}\)+\(\dfrac{1}{\left(x+1\right)\left(x+2\right)}\)+\(\dfrac{1}{\left(x+2\right)\left(x+3\right)}\)+\(\dfrac{1}{\left(x+3\right)\left(x+4\right)}\)+\(\dfrac{1}{\left(x+4\right)\left(x+5\right)}\)+\(\dfrac{1}{x+5}\) = \(\dfrac{1}{x}\)-\(\dfrac{1}{x+1}\)+\(\dfrac{1}{x+1}\)-\(\dfrac{1}{x+2}\)+\(\dfrac{1}{x+2}\)-...... -\(\dfrac{1}{x+5}\)+\(\dfrac{1}{x+5}\)=\(\dfrac{1}{x}\)
Vậy tổng trên bằng \(\dfrac{1}{x}\)
Bài 3: (SBT/24):
a. \(\dfrac{5x+3}{x-2}\)=\(\dfrac{5x^2+13x+6}{x^2-4}\)
(5x+3) . (x2-4) = 5x3-20x+3x3-12
(x-2) . (5x2+13x+6) = 5x3+13x2+6x-10x2-26x-12 = 5x3-20x+3x2-12
=> (5x+3) (x2-4) = (x-2) (5x2+13x+6)
Vậy \(\dfrac{5x+3}{x-2}\)=\(\dfrac{5x^2+13x+6}{x^2-4}\)(đẳng thức đúng)
b. \(\dfrac{x+1}{x+3}\)=\(\dfrac{x^2+3}{x^2+6x+9}\)
(x+1) . (x2+6x+9) = x3+6x2+9x+x2+6x+9 = x3+7x2+15x+9
(x+3) . (x2+3) = x3+3x+3x2+9
=> (x+1) (x2+6x+9) ≠ (x+3) (x2+3)
Vậy \(\dfrac{x+1}{x+3}\)≠\(\dfrac{x^2+3}{x^2+6x+9}\)(đẳng thức sai)
Chữa lại: \(\dfrac{x+1}{x+3}\)=\(\dfrac{x^2+3}{x^{2_{ }}+6x+9}\)
c. \(\dfrac{x^2-2}{x^2-1}\)=\(\dfrac{x+2}{x+1}\)
(x2-2) . (x+1) = x3+x2-2x-2
(x2-1) . (x+2) = x3+2x2-x-2
=> (x2-2) (x+1) ≠ (x2-1) (x+2)
Vậy \(\dfrac{x^2-2}{x^2-1}\)≠\(\dfrac{x+2}{x+1}\)(đẳng thức sai)
Chữa lại: \(\dfrac{x^2+x-2}{x^2-1}\)=\(\dfrac{x+2}{x+1}\)
d. \(\dfrac{2x^2-5x+3}{x^2+3x-4}\)=\(\dfrac{2x^2-x-3}{x^2+5x+4}\)
(2x2-5x+3) . (x2+5x+4) = 2x4+10x3+8x2-5x3-25x2-20x+3x2+15x+12
= 2x4+5x3-14x2-5x+12
(x2+3x-4) . (2x2-x-3) = 2x4-x3-3x2+6x3-3x2-9x-8x2+4x+12
= 2x4+5x3-14x2-5x+12
=> (2x2-5x+3) (x2+5x+4) = (x2+3x-4) (2x2-x-3)
Vậy \(\dfrac{2x^2-5x+3}{x^2+3x-4}\)=\(\dfrac{2x^2-x-3}{x^2+5x+4}\)
tính chất quan trọng phần thức với
\(\dfrac{a}{b}=\dfrac{c}{d}\Rightarrow ad=bc\Rightarrow c=\dfrac{ad}{b}\)áp vào
\(\dfrac{x^5-1}{x^2-1}=\dfrac{A}{x+1}\Rightarrow A=\dfrac{\left(x^5-1\right)\left(x+1\right)}{x^2-1}\) {x khác +-1}
\(A=\dfrac{\left(x^5-1\right)\left(x+1\right)}{\left(x-1\right)\left(x+1\right)}=\dfrac{\left[\left(x-1\right)\left(x^4+x^3+x^2+x+1\right)\right]\left(x+1\right)}{\left(x-1\right)\left(x+1\right)}=\left(x^4+x^3+x^2+x+1\right)\)
Vậy đa thức cần điền là
\(A=\left(x^4+x^3+x^2+x+1\right)\)
Gọi phân thức cần tìm là \(\dfrac{a}{b}\)
Theo đề bài ta có :
\(\dfrac{x}{x+1}:\dfrac{x+2}{x+1}:\dfrac{x+3}{x+2}:\dfrac{x+4}{x+3}:\dfrac{x+5}{x+4}:\dfrac{a}{b}=1\)
\(\Leftrightarrow\dfrac{x}{x+1}\cdot\dfrac{x+1}{x+2}\cdot\dfrac{x+2}{x+3}\cdot\dfrac{x+3}{x+4}\cdot\dfrac{x+4}{x+5}\cdot\dfrac{b}{a}=1\)
\(\Leftrightarrow\dfrac{x}{x+5}\cdot\dfrac{b}{a}=1\)
\(\Rightarrow\dfrac{b}{a}=\dfrac{x+5}{x}\)
\(\Rightarrow\dfrac{a}{b}=\dfrac{x}{x+5}\)
Vậy phân thức cần tìm là \(\dfrac{x}{x+5}\)