Bài tập file word mức độ vận dụng bài 23: Phép cộng và phép trừ phân thức đại số

Câu 1:  

a) $ \frac{a^{2}}{a+b}+\frac{b^{2}}{b+c}+\frac{c^{2}}{c+a}-\frac{a^{2}}{c+a}-\frac{b^{2}}{a+b}-\frac{c^{2}}{b+c}$

$=\frac{a^{2}-b^{2}}{a+b}+\frac{b^{2}-c^{2}}{b+c}+\frac{c^{2}-a^{2}}{c+a}$

$=a-b+b-c+c-a=0$

Vậy  $\frac{a^{2}}{a+b}+\frac{b^{2}}{b+c}+\frac{c^{2}}{c+a}=\frac{a^{2}}{c+a}+\frac{b^{2}}{a+b}+\frac{c^{2}}{b+c}$

Câu 2:

a) $\frac{4x^{2}-(x-3)^{2}}{9(x^{2}-1)}-\frac{x^{2}-9}{(2x+3)^{2}-x^{2}}+\frac{(2x-3)^{2}-x^{2}}{4x^{2}-(x+3)^{2}}$

$=\frac{(2x-x+3)(2x+x-3)}{9(x-1)(x+1)}-\frac{(x-3)(x+3)}{(2x+3-x)(2x+3+x)}+\frac{(2x-3-x)(2x-3+x)}{(2x-x-3)(2x+x+3)}$

$=\frac{3(x+3)(x-1)}{9(x-1)(x+1)}-\frac{(x-3)(x+3)}{3(x+3)(x+1)}+\frac{3(x-3)(x-1)}{3(x-3)(x+1)}$

$=\frac{x+3}{3(x+1)}-\frac{x-3}{3(x+1)}+\frac{3(x-1)}{3(x+1)}=\frac{x+3-x+3+3x-3}{3(x+1)}=\frac{3x+3}{3x+3}=1$

b) $\frac{y-z}{(x-y)(x-z)}+\frac{z-x}{(y-z)(y-x)}+\frac{x-y}{(z-x)(z-y)}$

$=\frac{(x-z)-(x-y)}{(x-y)(x-z)}+\frac{(y-x)-(y-z)}{(y-z)(y-x)}+\frac{(z-y)-(z-x)}{(z-x)(z-y)}$

$=\frac{1}{x-y}-\frac{1}{x-z}+\frac{1}{y-z}-\frac{1}{y-x}+\frac{1}{z-x}-\frac{1}{z-y}$

$=\frac{1}{x-y}+\frac{1}{z-x}+\frac{1}{y-z}+\frac{1}{x-y}+\frac{1}{z-x}+\frac{1}{y-z}$

$=\frac{2}{x-y}+\frac{2}{y-z}+\frac{2}{z-x}$

Câu 3: 

Xét tử thức 

$a^{3}-b^{3}+c^{3}+3abc$

$=a^{3}-3ab(a-b)-b^{3}+c^{3}+3ab(a-b)+3abc$

$=(a-b)^{3}+c^{3}+3ab(a-b+c)$

$=(a-b+c)\left [ (a-b)^{2}+c^{2}-c(a-b)+3ab \right ]$

$=(a-b+c)(a^{2}+b^{2}+c^{2}+ab+bc-ca)$

$=\frac{1}{2}(a-b+c)\left [ (a+b)^{2}+(b+c)^{2}+(c-a)^{2} \right ]$

Vậy

$D=\frac{a^{3}-b^{3}+c^{3}+3abc}{(a+b)^{2}+(b+c)^{2}+(c-a)^{2}}$

$=\frac{\frac{1}{2}(a-b+c)\left [ (a+b)^{2}+(b+c)^{2}+(c-a)^{2} \right ]}{(a+b)^{2}+(b+c)^{2}+(c-a)^{2}}$

$=\frac{1}{2}(a-b+c)$