I will now teach you in detail **how to multiply and divide algebraic fractions**, with step-by-step exercises.

I will teach you the tricks you need to perform the multiplication and division of algebraic fractions in the simplest way possible.

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## Multiplication of algebraic fractions. Exercise resolved step by step.

The algebraic fractions are multiplied just like the numerical fractions, that is, they are multiplied in line: numerator by numerator and denominator by denominator, only in this case, instead of numbers we have polynomials:

There is also another small difference (although it is only a recommendation) that I will explain to you:

In the multiplication of numerical fractions, the numbers in line are multiplied and at the end the fraction is simplified. With algebraic fractions, we can do the same, but the operations would be too complicated.

So, what I recommend is that before multiplying, we break down the polynomials and eliminate the factors that are repeated in the numerator and denominator, that is, simplify before multiplying.

Once we have eliminated all the repeated factors, we can now multiply both the numerator and the denominator to show it in the result. I mean, we multiply in the end.

Let’s take a step-by-step example to make what I just said clear to you.

We have the following multiplication of algebraic fractions:

Being a multiplication of fractions, we multiply in line, that is, numerator by numerator and denominator by denominator, but being polynomials, we only leave it indicated, we do not multiply them:

Before we multiply, we are going to decompose the polynomials that can decompose. We begin with the polynomial corresponding to the numerator of the first fraction:

We also decompose the polynomial of the denominator of the first fraction:

The other two polynomials cannot be broken down, as they are already grade 1.

We replace the polynomials by their corresponding decompositions:

Now we simplify the algebraic fraction, eliminating the factors that are repeated in the numerator and in the denominator:

And we’re left with it:

That we multiply to get the final result:

If we had multiplied at the beginning, in the end we would have had two polynomials of greater degree, which would have been much more difficult to factor.

By following this procedure, we arrive at the result much more directly.

Let’s now look at the division of algebraic fractions.

## Division of algebraic fractions. Exercise resolved step by step

The division of algebraic fractions is also done in the same way as a division of numerical fractions, i.e. it is multiplied in a cross:

As in the case of multiplication, it is also advisable to leave the indicated multiplication and factor the polynomials before multiplying, in order to arrive at the simplified result in a more direct way.

Let’s look at it with another example:

Being a division of algebraic fractions, we multiply the cross polynomials and leave it indicated (without multiplying the polynomials):

Now we decompose the polynomials:

One of the polynomials of the numerator cannot be decomposed although it is of degree 2, since its function has no root solutions.

We replace each polynomial by its decomposition:

We eliminate the factors that are repeated in the numerator and in the denominator:

And we’re left with it:

Finally, we multiply the factors we have in the denominator to obtain the final result:

In the case of division, by breaking down the polynomials before multiplying them into crosses, we also get a directly simplified result.

## Exercises solved

Solve the following additions and subtractions of algebraic fractions: