Factorization and algebraic fractions - Examples, Exercises and Solutions

Algebraic fractions are fractions with variables.

Ways to factor algebraic fractions:

  1. We will find the most appropriate common factor to extract.
  2. 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.
  3. If the formulas for abbreviated multiplication cannot be used, we will proceed to factorize with trinomials.
  4. 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).

Suggested Topics to Practice in Advance

  1. Factoring using contracted multiplication
  2. Factorization
  3. Extracting the common factor in parentheses
  4. Factorization: Common factor extraction
  5. Factoring Trinomials

Practice Factorization and algebraic fractions

Exercise #1

Determine if the simplification shown below is correct:

778=8 \frac{7}{7\cdot8}=8

Video Solution

Step-by-Step Solution

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

77×18 \frac{7}{7}\times\frac{1}{8}

We simplify:

1×18=18 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:

5883=53 \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:

88×53 \frac{8}{8}\times\frac{5}{3}

We simplify:

1×53=53 1\times\frac{5}{3}=\frac{5}{3}

Answer

Correct

Exercise #3

Determine if the simplification below is correct:

6363=1 \frac{6\cdot3}{6\cdot3}=1

Video Solution

Step-by-Step Solution

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

=?11=!1 \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.

Answer

Correct

Exercise #4

Complete the corresponding expression for the denominator

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

Video Solution

Step-by-Step Solution

We use the formula:

xy=zwxy=zy \frac{x}{y}=\frac{z}{w}\xrightarrow{}x\cdot y=z\cdot y

We convert the 8 into a fraction, and multiply

16ab?=81 \frac{16ab}{?}=\frac{8}{1}

16ab×1=8a 16ab\times1=8a

16ab=8a 16ab=8a

We divide both sides by 8a:

16ab8a=8a8a \frac{16ab}{8a}=\frac{8a}{8a}

2b 2b

Answer

2b 2b

Exercise #5

Determine if the simplification below is correct:

3773=0 \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:

77×33=1×1=1 \frac{7}{7}\times\frac{3}{3}=1\times1=1

Therefore, the simplification described is false.

Answer

Incorrect

Exercise #1

Determine if the simplification below is correct:

484=18 \frac{4\cdot8}{4}=\frac{1}{8}

Video Solution

Step-by-Step Solution

We will divide the fraction exercise into two multiplication exercises:

44×81= \frac{4}{4}\times\frac{8}{1}=

We simplify:

1×81=8 1\times\frac{8}{1}=8

Therefore, the described simplification is false.

Answer

Incorrect

Exercise #2

Determine if the simplification below is correct:

3xx+3=0 \frac{3-x}{-x+3}=0

Video Solution

Step-by-Step Solution

zxx+z=1 \frac{z-x}{-x+z}=1

Answer

Incorrect

Exercise #3

Determine if the simplification described below is correct:

x+6y+6=xy \frac{x+6}{y+6}=\frac{x}{y}

Video Solution

Step-by-Step Solution

We use the formula:

x+zy+z=x+zy+z \frac{x+z}{y+z}=\frac{x+z}{y+z}

x+6y+6=x+6y+6 \frac{x+6}{y+6}=\frac{x+6}{y+6}

Therefore, the simplification described is incorrect.

Answer

Incorrect

Exercise #4

Determine if the simplification below is correct:

3483=12 \frac{3\cdot4}{8\cdot3}=\frac{1}{2}

Video Solution

Step-by-Step Solution

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

First let's consider the fact that the number 8 is a multiple of the number 4:

8=24 8=2\cdot4
Therefore, we will return to the problem in question and present the number 8 as a multiple of the number 4, then we will simplify the fraction:

3483=?1234243=?122=?1212=!12 \frac{3\cdot4}{\underline{8}\cdot3}\stackrel{?}{= }\frac{1}{2}\\ \downarrow\\ \frac{3\cdot4}{\underline{2\cdot4}\cdot3}\stackrel{?}{= }\frac{1}{2}\\ \downarrow\\ \frac{\textcolor{blue}{\not{3}}\cdot\textcolor{red}{\not{4}}}{2\cdot\textcolor{red}{\not{4}}\cdot\textcolor{blue}{\not{3}}}\stackrel{?}{= }\frac{1}{2} \\ \downarrow\\ \frac{1}{2}\stackrel{!}{= }\frac{1}{2}
Therefore, the described simplification is correct.

That is, the correct answer is A.

Answer

True

Exercise #5

Select the field of application of the following fraction:

713+x \frac{7}{13+x}

Video Solution

Answer

x13 x\neq-13

Exercise #1

Select the field of application of the following fraction:

82+x \frac{8}{-2+x}

Video Solution

Answer

x2 x\neq2

Exercise #2

Complete the corresponding expression for the denominator

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

Video Solution

Answer

12ab 12ab

Exercise #3

Complete the corresponding expression for the denominator

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

Video Solution

Answer

8a 8a

Exercise #4

Complete the corresponding expression for the denominator

27ab?=3ab \frac{27ab}{\text{?}}=3ab

Video Solution

Answer

9 9

Exercise #5

Complete the corresponding expression for the denominator

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

Video Solution

Answer

19b 19b

Topics learned in later sections

  1. Algebraic Fractions
  2. Simplifying Algebraic Fractions
  3. Addition and Subtraction of Algebraic Fractions
  4. Multiplication and Division of Algebraic Fractions
  5. Solving Equations by Factoring
  6. Uses of Factorization