More complex algebraic fractions often involve linear expressions in the denominator, such as \( x + 3 \) or \( 2x - 1 \). Simplifying these fractions may require factorising, finding common denominators and carefully adding or subtracting fractions.

 

As before, only factors can be cancelled, not terms.

 

 
Simplifying Algebraic Fractions by Cancelling Factors

If the numerator and denominator share a common factor, it can be cancelled.

 

For example:

$$
\frac{4x}{2x}
$$

 

Both the numerator and denominator have a factor of \( 2x \):

$$
= 2
$$

This is only valid if \( x \neq 0 \).

 

Another example:

$$
\frac{6(x + 2)}{3(x + 2)}
$$

 

Cancel the common factor \( x + 2 \):

$$
= 2
$$

This is only valid if \( x \neq -2 \).

 

 

Adding Algebraic Fractions with Linear Denominators

To add algebraic fractions with different linear denominators, a common denominator is required.

 

For example:

$$
\frac{x}{x + 3} + \frac{2}{x + 3}
$$

 

The denominators are already the same, so add the numerators:

$$
= \frac{x + 2}{x + 3}
$$

 

If the denominators are different, they must be made the same.

 

For example:

$$
\frac{1}{x} + \frac{3}{x + 1}
$$

 

The common denominator is:

$$
x(x + 1)
$$

 

Rewrite each fraction:

$$
\frac{x + 1}{x(x + 1)} + \frac{3x}{x(x + 1)}
$$

 

Add the numerators:

$$
= \frac{x + 1 + 3x}{x(x + 1)}
$$

$$
= \frac{4x + 1}{x(x + 1)}
$$

 

 
Subtracting Algebraic Fractions with Linear Denominators

Subtraction follows the same method as addition, but signs must be handled carefully.

 

For example:

$$
\frac{x}{x - 2} - \frac{1}{x - 2}
$$

 

The denominators are the same, so subtract the numerators:

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

 

For different denominators:

$$
\frac{2}{x} - \frac{1}{x - 3}
$$

 

The common denominator is:

$$
x(x - 3)
$$

 

Rewrite each fraction:

$$
\frac{2(x - 3)}{x(x - 3)} - \frac{x}{x(x - 3)}
$$

 

Subtract the numerators:

$$
= \frac{2x - 6 - x}{x(x - 3)}
$$

$$
= \frac{x - 6}{x(x - 3)}
$$

 

 
Important Restrictions

When simplifying algebraic fractions, certain values of the variable may be excluded. These are the values that make a denominator equal to zero.

 

For example, in the fraction:

$$
\frac{1}{x - 3}
$$

The value \( x = 3 \) is not allowed.

 

Always state or be aware of these restrictions when working with algebraic fractions.

 

 
Key Points to Remember

• A common denominator is required before adding or subtracting.
• Only cancel common factors, not terms.
• Simplify fully after adding or subtracting.
• Check for values that make denominators zero.

 

Careful algebraic manipulation ensures complex fractions are simplified correctly and remain valid.