Simplification of algebraic fractions. Exercises resolved step by step.

Next I will show you how to simplify algebraic fractions and what are the equivalent algebraic fractions, two concepts that will be very helpful when performing operations with algebraic fractions.

What is an algebraic fraction

An algebraic fraction is the quotient between two polynomials:

simplification of algebraic fractions

Where P(x) and Q(x) are two polynomials.

For example:

simplify algebraic fractions

Value of an Algebraic Fraction

The value of an algebraic fraction is the value that the fraction takes when we replace the variables by a given number.

It is the same as calculating the numerical value of the polynomials that make up the fraction and then making their quotient.

Let’s see an example: Calculate the numerical value of the next algebraic fraction for the value of x that is indicated:

algebraic fraction simplification exercises

We replace the x with the 2 and calculate:

simplification of algebraic fractions exercises

What are the equivalent algebraic fractions

Two algebraic fractions are equivalent when they have the same numerical value.

Will these two fractions be equivalent?

simplifying algebraic fractions

Let’s calculate the numerical value for each of them.

The value of x that we have to take must be the same for both of us and must be one that does not result in a zero in the denominator.

We will calculate the numerical value of each fraction for x=2.

For the first algebraic fraction we calculated it in the previous section:

simplification of algebraic fractions solved exercises

We calculate the numerical value of the second algebraic fraction for x=2:

simplification exercises

Both values are the same, so the algebraic fractions are equivalent:

simplification of algebraic expressions exercises

In general, two algebraic fractions:

algebraic fraction exercises

are equivalent if the multiplication of the numerator of one by the denominator of the other is also the same (cross multiplication):

algebraic fractions exercises

And therefore, the algebraic fractions will also be the same:

factoring fractions resolved exercises

In this way, it is not necessary to calculate the numerical value of the fractions. Let’s check it out with the same two algebraic fractions from the previous example:

algebraic simplification

We multiply the numerator of the first by the denominator of the second:

algebraic expression simplification exercises

We operate and the result is:

simplify algebraic expressions solved exercises

We multiply the denominator of the first by the numerator of the second:

simplify algebraic fractions solved exercises

Whose result is:

algebraic simplifications

how to simplify algebraic fractions

In both cases we have obtained the same polynomial, so the fractions are equivalent:

simplifying algebraic fractions exercises

On the other hand, if we multiply the numerator and denominator of an algebraic fraction by the same polynomial, the resulting algebraic fraction is a fraction equivalent to the previous one.

For example, we have this algebraic fraction:

resolved simplification exercises

And multiply numerator and denominator by (x-3)

simplification exercises solved

We’ve got another fraction as a result. Let’s see if they’re really equivalent:

algebraic fractions examples

We multiply in crosses:

algebraic fraction simplification exercises

simplify algebraic fractions exercises

And we see that both multiplications result in the same polynomial, so they are equivalent.

How to simplify algebraic fractions

Let’s now look at how to simplify algebraic fractions.

In the same way that if you multiply an algebraic fraction by the same polynomial the numerator and the denominator you get another equivalent algebraic fraction, if we divide between the same polynomial, we will get another equivalent algebraic fraction, whose polynomials will have one degree less and therefore we will have simplified it.

Dividing the numerator and the denominator by the same polynomial is equivalent to eliminating the same polynomial from the numerator and the denominator, which will be in the form of a factor.

At the same time, removing the same polynomial from the numerator and the denominator is equivalent to multiplying by 1.

To simplify an algebraic fraction, we must first decompose the polynomials of the numerator and denominator. The factor or factors that are repeated above and below are those that we can eliminate.

For example: Simplify the following algebraic fraction:

exercises algebraic fractions

We decompose the numerator and the denominator:

algebraic simplification examples

We see that the factor (x+1), is repeated up and down, so we can eliminate it and it remains:

algebraic simplification exercises

That is a fraction whose polynomials are of lower degree than the original and is equivalent to the same.