Algebraic fractions are fractions with variables.

Algebraic fractions are fractions with variables.

- 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).

Complete the corresponding expression for the denominator

\( \frac{12ab}{?}=1 \)

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:**

$x+4$

Complete the corresponding expression for the denominator

$\frac{16ab}{?}=8a$

Using the formula:

$\frac{x}{y}=\frac{z}{w}\xrightarrow{}x\cdot y=z\cdot y$

We first convert the 8 into a fraction, and multiply

$\frac{16ab}{?}=\frac{8}{1}$

$16ab\times1=8a$

$16ab=8a$

We then divide both sides by 8a:

$\frac{16ab}{8a}=\frac{8a}{8a}$

$2b$

$2b$

Determine if the simplification below is correct:

$\frac{3\cdot7}{7\cdot3}=0$

We will divide the fraction exercise into two different multiplication exercises.

As this is a multiplication exercise, you can use the substitution property:

$\frac{7}{7}\times\frac{3}{3}=1\times1=1$

Therefore, the simplification described is false.

Incorrect

Determine if the simplification below is correct:

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

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.

Incorrect

Determine if the simplification below is correct:

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

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

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

We simplify:

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

Correct

Determine if the simplification below is correct:

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

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.

__Therefore, the correct answer is A.__

Correct

Test your knowledge

Question 1

Complete the corresponding expression for the denominator

\( \frac{16ab}{?}=2b \)

Question 2

Complete the corresponding expression for the denominator

\( \frac{16ab}{?}=8a \)

Question 3

Complete the corresponding expression for the denominator

\( \frac{19ab}{?}=a \)