# Algebraic Fractions

## Algebraic Fractions Revision

**Algebraic Fractions**

An **algebraic fraction** is a fraction involving algebraic expressions. You can manipulate them in the same way as normal fractions, however it is a little bit harder – therefore it is helpful to simplify the fractions first.

There are **3** skills you need to learn for algebraic fractions.

Make sure you are happy with the following topics before continuing.

**Skill 1: Simplifying Algebraic Fractions**

To **simplify** algebraic fractions, you need to factorise the denominator and numerator (if necessary), and then look for common factors in the numerator and denominator and cancel.

**Example:** Simplify the following fraction: \dfrac{8x+12}{4x^2 - 9}

Factorise the numerator and the denominator:

\dfrac{8x+12}{4x^2 - 9} = \dfrac{4(2x+3)}{(2x+3)(2x-3)}

Then, cancel any common factors:

\dfrac{\textcolor{red}{4}\cancel{(2x+3)}}{\cancel{(2x+3)} \textcolor{red}{(2x-3)}} = \textcolor{red}{\dfrac{4}{2x-3}}

**Skill 2: Multiplying and Dividing Algebraic Fractions**

To **multiply** algebraic fractions, you multiply the numerators together and multiply the denominators together – the same way you would for normal fractions.

To **divide** by an algebraic fraction, you flip the fraction and then multiply.

**Example:** Simplify the following expression: \textcolor{red}{\dfrac{x^2-2x-3}{2x}} \div \dfrac{\textcolor{limegreen}{x+1}}{\textcolor{blue}{x^2}}

**Step 1:** Factorise the numerator and denominator:

\textcolor{red}{\dfrac{x^2-2x-3}{2x}} \div \dfrac{\textcolor{limegreen}{x+1}}{\textcolor{blue}{x^2}} = \textcolor{red}{\dfrac{(x+1)(x-3)}{2x}} \div \dfrac{\textcolor{limegreen}{x+1}}{\textcolor{blue}{x^2}}

**Step 2:** Flip the second fraction and turn the divide into a multiply:

\textcolor{red}{\dfrac{(x+1)(x-3)}{2x}} \div \dfrac{\textcolor{limegreen}{x+1}}{\textcolor{blue}{x^2}} = \textcolor{red}{\dfrac{(x+1)(x-3)}{2x}} \times \dfrac{\textcolor{blue}{x^2}}{\textcolor{limegreen}{x+1}}

**Step 3:** Multiply the numerator together and the denominators together (combine into one fraction):

\textcolor{red}{\dfrac{(x+1)(x-3)}{2x}} \times \dfrac{\textcolor{blue}{x^2}}{\textcolor{limegreen}{x+1}} = \dfrac{x^2(x+1)(x-3)}{2x(x+1)}

**Step 4:** Cancel down any common factors:

\dfrac{x ^{\cancel{2}} \cancel{(x+1)}(x-3)}{2\cancel{x} \cancel{(x+1)}} = \dfrac{x(x-3)}{2} \,\,\, \left( = \dfrac{x^2 - 3x}{2} \right)

**Skill 3****: Adding and Subtracting Algebraic Fractions**

You use the same methods for normal fractions to **add** and **subtract** algebraic fractions.

**Example:** Simplify \dfrac{2x}{x+1} + \dfrac{5}{x^2} - \dfrac{1}{x}

**Step 1:** Find a common denominator – multiply the numerator and denominator of each fraction by the denominators of the other fractions:

\left( \dfrac{2x}{x+1} \times \textcolor{blue}{\dfrac{x^2}{x^2}} \times \textcolor{limegreen}{\dfrac{x}{x}} \right) + \left( \dfrac{5}{x^2} \times \textcolor{red}{\dfrac{x+1}{x+1}} \times \textcolor{limegreen}{\dfrac{x}{x}} \right) - \left( \dfrac{1}{x} \times \textcolor{red}{\dfrac{x+1}{x+1}} \times \textcolor{blue}{\dfrac{x^2}{x^2}} \right) = \dfrac{2x^4}{x^3(x+1)} + \dfrac{5x(x+1)}{x^3(x+1)} - \dfrac{x^2(x+1)}{x^3(x+1)}

**Step 2:** Now we can put everything over a common denominator, and then add the numerators together:

\dfrac{2x^4}{x^3(x+1)} + \dfrac{5x(x+1)}{x^3(x+1)} - \dfrac{x^2(x+1)}{x^3(x+1)} = \dfrac{2x^4 + 5x(x+1) - x^2(x+1)}{x^3(x+1)}

**Step 3:** Simplify the fraction as much as possible:

\dfrac{2x^4 + 5x(x+1) - x^2(x+1)}{x^3(x+1)} = \dfrac{2x^4 - x^3 + 4x^2 + 5x}{x^3(x+1)}

Notice that x is a common factor in the numerator and denominator, so the fraction can be cancelled further:

\dfrac{2x^{\cancel{4}} - x^{\cancel{3}} + 4x^{\cancel{2}} + 5\cancel{x}}{x^{\cancel{3}}(x+1)} = \dfrac{2x^3 - x^2 + 4x + 5}{x^2(x+1)}

**Example 1: Simplifying Algebraic Fractions**

Simplify the following:

**a)** \dfrac{27x^5y^3}{9x^4y}

**b)** \dfrac{2 + \dfrac{1}{3x}}{6x^2 + x}

**[4 marks]**

**a)** Here, you do not need to factorise, you just need to cancel down common factors:

\dfrac{\cancel{27}x^{\cancel{5}}y^{\cancel{3}}}{\cancel{9} x^{\cancel{4}} \cancel{y}} = 3xy^2

**b)** Here, you need to multiply the numerator and denominator by the same factor to cancel the fraction in the numerator:

\dfrac{2 + \dfrac{1}{3x}}{6x^2 + x} \times \dfrac{3x}{3x} = \dfrac{6x+1}{18x^3 + 3x^2}

Then, you can factorise and simplify as normal:

\dfrac{6x+1}{18x^3 + 3x^2} = \dfrac{\cancel{6x+1}}{3x^2 \cancel{(6x+1)}} = \dfrac{1}{3x^2}

**Example 2: Multiplying Algebraic Fractions**

Simplify the following algebraic fraction fully:

\dfrac{3x+6}{5xy} \times \dfrac{x}{x+2}

**[3 marks]**

Multiply the numerators together and the denominators together:

\dfrac{3x+6}{5xy} \times \dfrac{x}{x+2} = \dfrac{x(3x+6)}{5xy(x+2)}

Notice that 3x+6 = 3(x+2)

Then, simplify the fraction, by cancelling down any common factors:

\dfrac{x(3x+6)}{5xy(x+2)} = \dfrac{3\cancel{x} \cancel{(x+2)}}{5\cancel{x}y \cancel{(x+2)}} = \dfrac{3}{5y}

## Algebraic Fractions Example Questions

**Question 1:** Simplify the following fraction:

\dfrac{ax^2 + ay}{az}

**[2 marks]**

**Question 2:** Simplify the following fraction:

\dfrac{2 - \dfrac{1}{x^2}}{2x^2-1}

**[2 marks]**

**Question 3:** Simplify the following expression:

\dfrac{3x}{x+4} \div \dfrac{x^2}{2x+8}

**[3 marks]**

**Question 4:** Simplify the following expression:

\dfrac{4x}{x+1} - \dfrac{1}{2x}

**[3 marks]**

**Question 5:** Simplify the following expression:

\dfrac{x^2-x-12}{x+2} \times \dfrac{x^2 - 4}{x^2 + 3x}

**[3 marks]**

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