Equivalent fractions

An algebraic fraction is a division of the polynomial quotient.

Let's see some examples:

Example

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

$\frac{x^{4} + x^{3} + x - 1}{x^{3} + 2 x + 3}$

$\frac{x^{6}}{x - 2}$

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: $\frac{p (x)}{q (x)} and \frac{r (x)}{s (x)}$ two pairs of algebraic fractions, they will be equivalent if, and only if: $p (x) \cdot s (x) = r (x) \cdot q (x)$

Example

Let's see if this pair of algebraic fractions is equivalent: $\frac{x^{2} - 1}{x} and \frac{(x - 1)^{2}}{x}$ To verify, we will compute the cross products: $(x^{2} - 1) \cdot x = x^{3} - x$ $x \cdot (x - 1)^{2} = x \cdot (x^{2} - 2 x + 1) = x^{3} - 2 x^{2} + x$ They obviously are not equal. Therefore, the previous fractions are not equivalent.

Example

Let's see if this pair of algebraic fractions is equivalent: $\frac{x - 1}{x + 1} and \frac{(x - 1)^{2}}{x^{2} - 1}$ To verify, we will operate the cross products: $(x - 1) \cdot (x^{2} - 1) = x \cdot (x^{2} - 1) - 1 \cdot (x^{2} - 1) =$ $= x^{3} - x - x^{2} + 1 = x^{3} - x^{2} - x + 1$

$(x + 1) \cdot (x - 1)^{2} = (x + 1) \cdot (x^{2} - 2 x + 1) =$ $x \cdot (x^{2} - 2 x + 1) + 1 \cdot (x^{2} - 2 x + 1) = x^{3} - 2 x^{2} + x + x^{2} - 2 x + 1 =$ $= x^{3} - x^{2} - x + 1$ We can see that they are equal, and therefore, the previous fractions are equivalent.