πPractice factorization and algebraic fractions

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Factorization

Factorization and Algebraic Fractions

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Multiplication and Division Operations in Algebraic Fractions

When we want to multiply or divide algebraic fractions, we will use the same tools that we use for the multiplication or division of common fractions with some small differences.

Steps to carry out for the multiplication of algebraic fractions $1$:

Let's try to extract the common factor. This can be the variable or any free number.

How is the solution set found? We will make all the denominators we have equal to $0$ and find the solution. The solution set will be $X$: different from what causes our denominator to equal zero.

Let's simplify the fractions with determination.

Multiply numerator by numerator and denominator by denominator as in any fraction.

Steps to carry out for the division of algebraic fractions$2$:

We will convert the division exercise into a multiplication one, as we do with common fractions. How will we do it correctly? We will leave the first fraction as it is, change the division sign to a multiplication sign, and invert the fraction that appears after the sign. That is, numerator instead of denominator and denominator instead of numerator.

We will act according to the rules of multiplication of algebraic fractions:

Let's try to extract the common factor. This can be the unknown or any free number.

If this is not enough, we will factorize using short multiplication formulas and with trinomials.

Let's find the solution set.

How is the solution set found? We will make all the denominators we have equal to $0$ and find the solution. The solution set will be $X$: different from what causes our denominator to equal zero.

Let's simplify the fractions with determination.

Multiply numerator by numerator and denominator by denominator as in any fraction.

Let's look at an example of multiplying algebraic fractions

$\frac{x+2}{x+3}\times \frac{3x+9}{x^2-4}=$

Let's try to factorize by extracting the common factor and with the shortcut multiplication formulas, and we will obtain: $\frac{x+2}{x+3}\times \frac{3(x+3)}{(x-2)(x+2}=$

Let's find the solution set:

$xβ -3,2,-2$

Let's reduce the fractions and we will obtain:

$1\times \frac{3}{(x-2)}=$ Multiply and it will give us: $\frac{3}{x-2}$

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Let's convert the division exercise into a multiplication one:

$\frac{x^2-8x+15}{x^2-3x+2}\times \frac{x-1}{x^{2-9}}=$ Now, let's factor and we will get: $\frac{(x-5)(x-3)}{(x-2)(x-1)}\times \frac{x-1}{(x-3)(x+3)}=$ Let's find the solution set: $xβ 2,1,3,-3$

Let's simplify, we will get:

$\frac{x-5}{x-2}\times \frac{1}{x+3}$

Let's multiply and it will give us: $\frac{x-5}{(x-2)(x+3)}$

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Examples and exercises with solutions for multiplication and division of algebraic fractions

Exercise #1

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 #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 below is correct:

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

Video Solution

Step-by-Step Solution

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.

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.