An algebraic fraction is a fraction that contains at least one algebraic expression (with a variable) such as $3x$.

The expression can be in the numerator or the denominator or both.

An algebraic fraction is a fraction that contains at least one algebraic expression (with a variable) such as $3x$.

The expression can be in the numerator or the denominator or both.

We can simplify algebraic fractions only when there is a multiplication operation between the algebraic factors in the numerator and the denominator, and there are no addition or subtraction operations.**Steps to simplify algebraic fractions:**

- The first step โ

Attempt to factor out a common factor. - The second step โ

Attempt to simplify using special product formulas. - The third step โ

Attempt to factor by using a trinomial.

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**How do you reduce algebraic fractions?**

- We will find the common factor that is most beneficial for us to extract.
- If we do not find one, we will proceed to factorization using the formulas for shortened multiplication.
- If we cannot use the formulas for shortened multiplication, we will proceed to factorization using trinomials.
- We will simplify (only when there is multiplication between the terms unless the terms are in parentheses, in which case we will treat it as a single term).

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We will make all the denominators the same โ we will reach a common denominator.

We will use factorization according to the methods we have learned.

Steps of the operation:

- We will factor all the denominators.
- We will multiply each numerator by the number it needs so that its denominator reaches the common denominator.
- We will write the exercise with one denominator - the common denominator, and between the expressions in the numerators, we will keep the arithmetic operations as in the original exercise.
- After opening parentheses, we might encounter another expression that we need to factor. We will factor it and see if we can simplify.
- We will get a regular fraction and solve it.

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Steps to multiply algebraic fractions:

- Let's try to factor out a common factor.

The common factor can be our variable or any constant number. - If factoring out a common factor is not enough, we will reduce using the formulas for the product of sums or using trinomials.
- Let's find the domain of substitution:

We will set all the denominators we have to 0 and find the solutions.

The domain of substitution will be: x different from what makes the denominator zero. - Let's simplify the fractions.
- We will multiply numerator by numerator and denominator by denominator as in a regular fraction.

Steps for dividing algebraic fractions:

- We will turn the division exercise into a multiplication exercise in this way:

We will leave the first fraction as it is, change the division sign to a multiplication sign, and invert the fraction that comes after the division operation. That is, numerator in place of denominator and denominator in place of numerator. - We will follow the rules for multiplying algebraic fractions:
- We will try to factor out a common factor.

The common factor can be our variable or any free number. - If factoring out a common factor is not enough, we will decompose using the formulas for shortened multiplication and also using trinomials.
- We will find the domain of substitution:

We will set all the denominators we have to 0 and find the solutions.

The domain of substitution will be x different from what zeros the denominator. - We will simplify the fractions.
- We will multiply numerator by numerator and denominator by denominator as in a regular fraction.

- We will try to factor out a common factor.

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Select the field of application of the following fraction:

\( \frac{7}{13+x} \)

**Exercise:**

Simplify the following algebraic fraction:

$\frac{4x+2}{2x}$

Solution:

First, we factor out the common factor $4$ in the numerator and get:

$\frac{4(x+1)}{2x}$

Now we simplify the $4$ by $2$ and get:

$\frac{2(x+1)}{x}$

**Another exercise:**

Simplify the following algebraic fraction:

$\frac{x+10}{20}$

Solution:

The fraction cannot be simplified because $x$ is not involved in multiplication but in addition.

**Exercise:**

$\frac{1}{x^2-25}+\frac{1}{x^2-10x+25}=$

Let's factor all the denominators:

$\frac{1}{(x-5)(x+5)}+\frac{1}{(x-5)^2}=$

It is advisable to write down the common denominator in front of us, so it will be easier to know what to multiply each numerator by:

$(x+5) (x-5)^2$

We will multiply each numerator by what it needs so that its denominator reaches the common denominator, write the exercise with one denominator and get:

$\frac{x-5+x+5}{(x+5)(x-5)^2}=$

Combine terms in the numerator and get

$\frac{2x}{(x+5)(x-5)^2}$

This is the final answer.

Test your knowledge

Question 1

Select the field of application of the following fraction:

\( \frac{8}{-2+x} \)

Question 2

Determine if the simplification below is correct:

\( \frac{6\cdot3}{6\cdot3}=1 \)

Question 3

Determine if the simplification below is correct:

\( \frac{5\cdot8}{8\cdot3}=\frac{5}{3} \)