An algebraic fraction is a division of the polynomial quotient.
Let's see some examples:
$$\dfrac{x^2-3}{x+1}$$
$$\dfrac{x^4+x^3+x-1}{x^3+2x+3}$$
$$\dfrac{x^6}{x-2}$$
In a similar way as with fractions, we can define two algebraic fractions as equivalent if its cross product is equal. That is, if we have: $$$\dfrac{p(x)}{q(x)} \ \mbox{and} \ \dfrac{r(x)}{s(x)}$$$ two pairs of algebraic fractions, they will be equivalent if, and only if: $$$p(x)\cdot s(x)=r(x)\cdot q(x)$$$
Let's see if this pair of algebraic fractions is equivalent: $$$\dfrac{x^2-1}{x} \ \mbox{and} \ \dfrac{(x-1)^2}{x}$$$ To verify, we will compute the cross products: $$$(x^2-1)\cdot x=x^3-x$$$ $$$x\cdot(x-1)^2=x\cdot(x^2-2x+1)=x^3-2x^2+x$$$ They obviously are not equal. Therefore, the previous fractions are not equivalent.
Let's see if this pair of algebraic fractions is equivalent: $$$\dfrac{x-1}{x+1} \ \mbox{and} \ \dfrac{(x-1)^2}{x^2-1}$$$ To verify, we will operate the cross products: $$$(x-1)\cdot (x^2-1)=x\cdot(x^2-1)-1\cdot(x^2-1)=$$$ $$$=x^3-x-x^2+1=x^3-x^2-x+1$$$
$$$(x+1)\cdot(x-1)^2=(x+1)\cdot(x^2-2x+1)=$$$ $$$x\cdot (x^2-2x+1)+1\cdot(x^2-2x+1)=x^3-2x^2+x+x^2-2x+1=$$$ $$$=x^3-x^2-x+1$$$ We can see that they are equal, and therefore, the previous fractions are equivalent.