Equivalent Algebraic Fractions

Two algebraic fractions are considered equivalent if they yield the same quotient and remainder: $$ \frac{A}{B} \sim \frac{C}{D} $$, where A, B, C, and D are polynomials. In this case, their cross products are equal: $$ A \cdot D = C \cdot B $$

In other words, the product of the numerator polynomial A of the first fraction and the denominator polynomial D of the second fraction must match the product of the numerator polynomial C of the second fraction and the denominator polynomial B of the first fraction.

$$ A \cdot D = C \cdot B $$

If the cross product comes out equal, then the two algebraic fractions A:B and C:D have the same quotient and remainder.

$$ \frac{A}{B} \sim \frac{C}{D} $$

Therefore, they are equivalent algebraic fractions.

Note. This is essentially the same principle you’ve already seen with numerical equivalent fractions, but here the terms are polynomials rather than plain numbers. The idea of equivalence between fractions, however, remains unchanged.

A practical example

Consider the following two algebraic fractions:

$$ \frac{2}{a+b} $$

$$ \frac{2a+2b}{(a+b)^2} $$

We want to check if these two fractions are equivalent.

To do so, let’s compute their cross product and see if the results match:

$$ 2 \cdot (a+b)^2 = (2a+2b) \cdot (a+b) $$

$$ 2 \cdot (a^2+2ab+b^2) = 2a \cdot (a+b) + 2b \cdot (a+b) $$

$$ 2a^2+4ab+2b^2 = 2a^2+2ab + 2ab + 2b^2 $$

$$ 2a^2+4ab+2b^2 = 2a^2+4ab+ 2b^2 $$

The cross products are indeed equal.

So, the two fractions are equivalent:

$$ \frac{2}{a+b} \sim \frac{2a+2b}{(a+b)^2} $$

Note. This means that if we assign any values to "a" and "b" that satisfy the conditions for the fractions to exist, both fractions will evaluate to the same quotient. For instance, let’s set a=2 and b=3: $$ \frac{2}{a+b} = \frac{2a+2b}{(a+b)^2} $$ $$ \frac{2}{2+3} = \frac{2(2)+2(3)}{(2+3)^2} $$ $$ \frac{2}{5} = \frac{4+6}{5^2} $$ $$ \frac{2}{5} = \frac{10}{25} $$ $$ \frac{2}{5} = \frac{2}{5} $$ As you can see, the quotient comes out the same in both cases.

Proof

Now let’s take two algebraic fractions with the same quotient and remainder:

$$ \frac{A}{B} = \frac{C}{D} $$

By the invariant property of equations, we multiply both sides by the polynomial B and simplify:

$$ \frac{A}{B} \cdot B = \frac{C}{D} \cdot B $$

$$ A = \frac{C}{D} \cdot B $$

Applying the invariant property again, this time multiplying through by the polynomial D, we get:

$$ A \cdot D = \frac{C}{D} \cdot B \cdot D $$

$$ A \cdot D = C \cdot B $$

So their cross products are equal.

Therefore, two algebraic fractions are equivalent precisely when their cross products are equal.

And so on.