Factoring Algebraic Fractions

Algebraic fractions are fractions with variables.

Ways to factor algebraic fractions:

We will find the most appropriate common factor to extract.
If we do not see a common factor that we can extract, we will move on to factorization with formulas for abbreviated multiplication as we have studied.
If the formulas for abbreviated multiplication cannot be used, we will proceed to factorize with trinomials.
We will reduce according to the rules of reduction (we can only reduce when there is multiplication between the terms unless they are within parentheses, in which case, we will consider them independent terms).

Detailed explanation

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Test yourself on factorization and algebraic fractions!

Select the field of application of the following fraction:

$\frac{x}{16}$

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Observe, you can factorize every expression included in your fraction separately in any way you desire and, in the end, you will arrive at the factorized expression.

Let's see an example of factoring algebraic fractions:
$\frac{x^2+7x+12}{x+3}=$

As you can see, in this fraction only the numerator can be factored.
We will factor it and obtain:
$\frac{(x+4)(x+3)}{(x+3)}=$
Now, we can reduce in the following way and we will obtain:

$x+4$

Examples and exercises with solutions on factoring algebraic fractions

Exercise #1

Select the domain of the following fraction:

$\frac{8+x}{5}$

Video Solution

Step-by-Step Solution

The domain depends on the denominator and we can see that there is no variable in the denominator.

Therefore, the domain is all numbers.

Answer

All numbers

Exercise #2

Select the the domain of the following fraction:

$\frac{6}{x}$

Video Solution

Step-by-Step Solution

The domain of a fraction depends on the denominator.

Since you cannot divide by zero, the denominator of a fraction cannot equal zero.

Therefore, for the fraction $\frac{6}{x}$ , the domain is "All numbers except 0," since the denominator cannot equal zero.

In other words, the domain is:

$x\ne0$