Algebraic Fractions Revision Notes for HSC SSCE Mathematics Advanced

Algebraic Fractions

What is an algebraic fraction?

An algebraic fraction is a fraction that contains algebraic expressions (like $x$ , $x^2$ , or $x + 1$ ) in either the numerator (top), the denominator (bottom), or both parts of the fraction.

For example: $\frac{x^2 - 1}{x - 1}$ is an algebraic fraction because both the numerator and denominator contain algebraic terms.

The golden rule

chatImportant

When working with algebraic fractions, always treat them exactly like ordinary numerical fractions. This fundamental principle means you should:

Factorise first - break down expressions into their factors before doing anything else
Cancel common factors - remove matching factors from numerator and denominator
Then perform operations - multiply, divide, add or subtract

Following this order prevents mistakes and simplifies your working considerably.

Simplifying algebraic fractions

Simplifying algebraic fractions means reducing them to their simplest form by removing common factors from the numerator and denominator.

Method:

Factorise both the numerator and the denominator completely
Cancel any common factors that appear in both
Write your simplified answer

lightbulbExample

Worked Example 1: Simplifying an Algebraic Fraction

Simplify: $\frac{x^2 - 1}{x - 1}$

Step 1: Factorise the numerator

The numerator $x^2 - 1$ is a difference of two squares, so it factorises to:

$x^2 - 1 = (x - 1)(x + 1)$

Step 2: Cancel the common factor

Now we can write:

$\frac{(x - 1)(x + 1)}{x - 1}$

The factor $(x - 1)$ appears in both numerator and denominator, so we can cancel it.

Answer:

$= x + 1$

Important warning about cancelling

chatImportant

You can only cancel factors, never individual terms. This is a critical rule that many students get wrong.

For example, you cannot cancel the $x$ in $\frac{x + 2}{x + 5}$ because $x$ is a term, not a factor. The $x$ is being added to other numbers, not multiplied.

You can only cancel when the numerator and denominator are fully factorised and you're removing a complete factor that appears in both.

Multiplying algebraic fractions

Multiplication of algebraic fractions follows the same pattern as numerical fractions.

Rule:

Multiply the numerators together and multiply the denominators together:

$\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}$

lightbulbExample

Worked Example 2: Multiplying Algebraic Fractions

Simplify: $\frac{2x}{3} \times \frac{9}{x}$

Step 1: Multiply straight across

Multiply the numerators: $2x \times 9 = 18x$

Multiply the denominators: $3 \times x = 3x$

This gives:

$= \frac{18x}{3x}$

Step 2: Cancel common factors

Both numerator and denominator can be divided by $3$ : $18 \div 3 = 6$

The $x$ cancels completely.

Answer:

$= 6$

infoNote

Exam tip:

Always look for opportunities to cancel before multiplying if possible. This reduces the size of numbers you're working with and minimises calculation errors. For instance, in the example above, you could spot that $9$ and $3$ share a common factor of $3$ , and that $x$ appears in both, allowing you to cancel before multiplying.

Dividing algebraic fractions

Division of algebraic fractions uses the "flip and multiply" rule.

Rule:

To divide by a fraction, flip the second fraction (find its reciprocal) and then multiply:

$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$

lightbulbExample

Worked Example 3: Dividing Algebraic Fractions

Simplify: $\frac{x}{4} \div \frac{3}{x}$

Step 1: Flip the second fraction

The second fraction $\frac{3}{x}$ becomes $\frac{x}{3}$ when flipped.

Now we multiply:

$= \frac{x}{4} \times \frac{x}{3}$

Step 2: Multiply

Multiply numerators: $x \times x = x^2$

Multiply denominators: $4 \times 3 = 12$

Answer:

$= \frac{x^2}{12}$

Adding and subtracting algebraic fractions

When adding or subtracting algebraic fractions, you need a common denominator, just like with numerical fractions.

When denominators are the same

If the fractions already have the same denominator, simply add or subtract the numerators and keep the denominator.

lightbulbExample

Worked Example 4: Adding with Same Denominators

Simplify: $\frac{1}{x} + \frac{2}{x}$

Step 1: Same denominator

Since both fractions have denominator $x$ , we can add the numerators:

$= \frac{1 + 2}{x}$

Answer:

$= \frac{3}{x}$

When denominators are different

If the denominators are different, you must find a common denominator first, then rewrite each fraction before combining.

lightbulbExample

Worked Example 5: Adding with Different Denominators

Simplify: $\frac{1}{x} + \frac{1}{x + 1}$

Step 1: Find the common denominator

The common denominator is $x(x + 1)$ - we multiply the two different denominators together.

Step 2: Rewrite each fraction

To convert $\frac{1}{x}$ to have denominator $x(x + 1)$ , multiply top and bottom by $(x + 1)$ :

$\frac{1}{x} = \frac{x + 1}{x(x + 1)}$

To convert $\frac{1}{x + 1}$ to have denominator $x(x + 1)$ , multiply top and bottom by $x$ :

$\frac{1}{x + 1} = \frac{x}{x(x + 1)}$

So we have:

$= \frac{x + 1}{x(x + 1)} + \frac{x}{x(x + 1)}$

Step 3: Add the numerators

Now both fractions have the same denominator, so add the numerators:

$= \frac{(x + 1) + x}{x(x + 1)}$

$= \frac{2x + 1}{x(x + 1)}$

Answer:

$\frac{2x + 1}{x(x + 1)}$

Common mistakes to avoid

chatImportant

Watch out for these common errors:

Cancelling terms instead of factors - Remember you can only cancel complete factors, not individual terms being added or subtracted
Forgetting to find a common denominator - When adding or subtracting, you must have the same denominator first
Not factorising first - Always factorise before attempting to simplify or cancel
Errors when flipping fractions in division - Make sure you flip the second fraction only, not the first one

Exam tips

infoNote

Always:

Factorise first before attempting any simplification

For addition and subtraction:

Write everything over a common denominator before combining the numerators

For division:

Flip the second fraction, then multiply

Final check:

After completing your working, check if further simplification is possible

Remember!

bookmarkSummary

Key Points to Remember:

Simplifying - factorise both numerator and denominator, then cancel common factors
Multiplying - multiply straight across (numerator with numerator, denominator with denominator)
Dividing - flip the second fraction and multiply
Adding/subtracting - find a common denominator first, then combine numerators
Always treat algebraic fractions exactly like numerical fractions

Algebraic Fractions (HSC SSCE Mathematics Advanced): Revision Notes