Simplifying Algebraic Fractions

Understanding Algebraic Fractions

An algebraic fraction is a fraction where the numerator and/or denominator contain algebraic expressions rather than just numbers. For example,

$\frac{3x}{2}$, $\frac{x^2 + 5x}{x - 1}$, or $\frac{2a + 3}{4b}$ are algebraic fractions.

The numerator is the expression above the fraction line, and the denominator is the expression below it. Unlike numerical fractions, algebraic fractions involve variables and require algebraic manipulation to simplify.

Simplifying Algebraic Fractions

To simplify an algebraic fraction, the goal is to reduce it to its simplest form by:

• Factorising both numerator and denominator fully
• Cancelling any common factors that appear in both numerator and denominator
• Writing the simplified fraction clearly

The key is to only cancel factors, not terms that are added or subtracted. For example, in $\frac{x+3}{x+5}$, you cannot cancel the $x$ because it is part of a sum, not a factor.

For instance, consider simplifying $\frac{6x}{9x^2}$. First, factorise both numerator and denominator:

$\frac{6x}{9x^2} = \frac{2 \times 3 \times x}{3 \times 3 \times x \times x} = \frac{2 \times \text{(cancelled 3)} \times \text{(cancelled x)}}{3 \times \text{(cancelled 3)} \times \text{(cancelled x)} \times x} = \frac{2}{3x}$

So the simplified form is $\frac{2}{3x}$.

Common Factorisation Techniques

Factorising is essential for simplifying algebraic fractions. The main techniques used here include:

Factorising Single Terms

Look for common numerical and variable factors in each term. For example,

Factorise $4x^2 - 8x$:

$$4x^2 - 8x = 4x(x - 2)$$

Factorising Quadratics

Quadratic expressions of the form $ax^2 + bx + c$ can often be factorised into two binomials:

Example: Factorise $x^2 + 5x + 6$

$$x^2 + 5x + 6 = (x + 2)(x + 3)$$

Using Difference of Squares

Expressions like $a^2 - b^2$ factorise as $(a - b)(a + b)$. For example:

$$x^2 - 9 = (x - 3)(x + 3)$$

These factorisation techniques help break down complex expressions into simpler factors, making it easier to cancel common factors in algebraic fractions.

For example, to simplify $\frac{x^2 - 9}{x^2 + 5x + 6}$, factorise numerator and denominator:

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

Cancel the common factor $(x + 3)$:

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

Restrictions on Simplification

When simplifying algebraic fractions, it is crucial to identify values of the variable that make the denominator zero, as these are not allowed (division by zero is undefined).

These values are called restrictions or domain restrictions. Even if a factor cancels during simplification, the original denominator’s zero-values must be stated as restrictions.

For example, simplify and state restrictions for:

$$\frac{x^2 - 4}{x^2 - x - 6}$$

Factor numerator and denominator:

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

Cancel $(x + 2)$, but note $x \neq -2$ and $x \neq 3$ because these values make the original denominator zero.

Simplified fraction:

$$\frac{x - 2}{x - 3}, \quad x \neq -2, \quad x \neq 3$$

Examples

Example: Simplify $\frac{2x^2 + 6x}{4x}$ and state any restrictions.

Factor numerator:

$$2x^2 + 6x = 2x(x + 3)$$

Rewrite the fraction:

$$\frac{2x(x + 3)}{4x}$$

Cancel common factor $2x$:

$$\frac{2x(x + 3)}{4x} = \frac{\cancel{2x}(x + 3)}{2 \times \cancel{x} \times 2} = \frac{x + 3}{2}$$

Restrictions: $x \neq 0$ (denominator cannot be zero).

For example, simplify $\frac{4x^2 - 12x}{8x}$ by factorising and cancelling common factors. Factor numerator: $4x^2 - 12x = 4x(x - 3)$. Rewrite fraction: $\frac{4x(x - 3)}{8x}$. Cancel common factor $4x$ to get $\frac{x - 3}{2}$. Restrictions: $x \neq 0$.