Giải bài tập 2 trang 42 sgk Toán 8 tập 1 CD

a. $\frac{4x+2}{4x-4}+\frac{3-6x}{6x-6}$ = $\frac{(4x+2).6}{4(x-1).6}+\frac{(3-6x).4}{6(x-1).4}$ = $\frac{24x+12}{24x-24}+\frac{12-24x}{24x-24}$

= $\frac{24x+12+12-24x}{24x-24}=\frac{24}{24(x-1)}=\frac{1}{x-1}$

b. $\frac{y}{2x^{2}-xy}+\frac{4x}{y^{2}-2xy}$ = $\frac{y(y^{2}-2xy)}{(2x^{2}-xy)(y^{2}-2xy)}+\frac{4x(2x^{2}-xy)}{(y^{2}-2xy)(2x^{2}-xy)}$

= $\frac{y^{3}-2x^{2}y+8x^{3}-4x^{2}y}{(y^{2}-2xy)(2x^{2}-xy)}$

= $\frac{y^{3}-6x^{2}y+8x^{3}}{(y^{2}-2xy)(2x^{2}-xy)}$

c. $\frac{x}{x-y}+\frac{y}{x+y}+\frac{2y^{2}}{x^{2}-y^{2}}$

= $\frac{x(x+y)}{(x-y)(x+y)}+\frac{y(x-y)}{(x+y)(x-y)}+\frac{2y^{2}}{(x-y)(x+y)}$

= $\frac{x^{2}+xy+xy-y^{2}+2y^{2}}{(x-y)(x+y)}$

= $\frac{x^{2}+2xy+y^{2}}{(x-y)(x+y)}$

= $\frac{(x+y)^{2}}{(x-y)(x+y)}=\frac{x+y}{x-y}$

d. $\frac{x^{2}+2}{x^{3}-1}+\frac{x}{x^{2}+x+1}+\frac{1}{1-x}$

= $\frac{x^{2}+2}{(x-1)(x^{2}+x+1)}+\frac{x(x-1)}{(x^{2}+x+1)(x-1)}-\frac{1.(x^{2}+x+1)}{(x^{2}+x+1)(x-1)}$

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

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

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