Select the domain of the following fraction: $$ \frac{8+x}{5} $$

All decomposed numbers except -5

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!

Complete the corresponding expression for the denominator

$\frac{12ab}{?}=1$

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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

Complete the corresponding expression for the denominator

$\frac{12ab}{?}=1$

Video Solution

Step-by-Step Solution

Let's examine the problem:

$\frac{12ab}{?}=1$ Now let's think logically, and remember the known fact that dividing any number by itself always yields the result 1,

Therefore, in order to get the result 1 from dividing two numbers, the only way is to divide the number by itself, meaning-

The missing expression in the denominator of the fraction on the left side is the complete expression that appears in the numerator of the same fraction:

$12ab$ .

Therefore- the correct answer is answer D.

Answer

$12ab$

Exercise #2

Determine if the simplification below is correct:

$\frac{4\cdot8}{4}=\frac{1}{8}$

Video Solution

Step-by-Step Solution

We will divide the fraction exercise into two multiplication exercises:

$\frac{4}{4}\times\frac{8}{1}=$

We simplify:

$1\times\frac{8}{1}=8$

Therefore, the described simplification is false.

Answer

Incorrect

Exercise #3

Determine if the simplification shown below is correct:

$\frac{7}{7\cdot8}=8$

Video Solution

Step-by-Step Solution

Let's consider the fraction and break it down into two multiplication exercises:

$\frac{7}{7}\times\frac{1}{8}$

We simplify:

$1\times\frac{1}{8}=\frac{1}{8}$

Therefore, the described simplification is false.

Answer

Incorrect

Exercise #4

Determine if the simplification below is correct:

$\frac{5\cdot8}{8\cdot3}=\frac{5}{3}$

Video Solution

Step-by-Step Solution

Let's consider the fraction and break it down into two multiplication exercises:

$\frac{8}{8}\times\frac{5}{3}$

We simplify:

$1\times\frac{5}{3}=\frac{5}{3}$

Answer

Correct

Exercise #5

Determine if the simplification below is correct:

$\frac{6\cdot3}{6\cdot3}=1$

Video Solution

Step-by-Step Solution

We simplify the expression on the left side of the approximate equality:

$\frac{\textcolor{red}{\not{6}}\cdot\textcolor{blue}{\not{3}}}{\textcolor{red}{\not{6}}\cdot\textcolor{blue}{\not{3}}}\stackrel{?}{= }1\\ \downarrow\\ 1\stackrel{!}{= }1$ therefore, the described simplification is correct.