Year 12 Algebra: Rational algebraic expressions

Rational algebraic expressions behave like fractions, but with the extra complication that variables can make a denominator zero. That means every simplification question has two jobs. First, simplify the expression using factorisation and fraction laws. Second, keep track of restrictions so you do not silently allow impossible values. Strong Year 12 work treats those restrictions as part of the answer, not as optional decoration added at the end.

Factor first, state denominator restrictions early, and only then cancel common factors.

Subtopic 1: Simplifying by factorisation

Many rational expressions look impossible until the numerator and denominator are factorised. Once that happens, common factors may appear and cancellation becomes legitimate. The important point is that you cancel factors, not individual terms. A common factor can be cancelled only when it multiplies the whole numerator and the whole denominator.

Worked example 1

Problem: Simplify

(x^{2} - 9) / (x^{2} - x - 6)

Factor the numerator: $x^{2} - 9 = (x - 3) (x + 3)$ .
Factor the denominator: $x^{2} - x - 6 = (x - 3) (x + 2)$ .
State the restrictions from the original denominator: $x \neq 3$ and $x \neq - 2$ .
Now cancel the common factor $(x - 3)$ .

Answer:

(x + 3) / (x + 2)

, with

x \neq 3, - 2

Subtopic 2: Adding and subtracting rational expressions

Rational expressions can only be added or subtracted once a common denominator has been built. Students often try to add numerators directly, which is the same kind of mistake as adding ordinary fractions without a shared denominator. The safest approach is to factor denominators when possible, identify the least common denominator, rewrite each fraction, then simplify the final numerator.

Worked example 2

Problem: Simplify

1 / x + 2 / (x + 1)

The least common denominator is $x (x + 1)$ .
Rewrite the first fraction: $1 / x = (x + 1) / (x (x + 1))$ .
Rewrite the second fraction: $2 / (x + 1) = 2 x / (x (x + 1))$ .
Add the numerators: $(x + 1 + 2 x) / (x (x + 1))$ .
Simplify: $(3 x + 1) / (x (x + 1))$ .
Restrictions are $x \neq 0$ and $x \neq - 1$ .

Answer:

(3 x + 1) / (x (x + 1))

, with

x \neq 0, - 1

Why restrictions stay from the original form

A common source of confusion is that restrictions come from the original expression, not from the simplified appearance after cancellation. In the first worked example, the factor $(x - 3)$ disappeared from the visible denominator, but $x = 3$ is still excluded because the original expression was undefined there. If you forget this point, you can write an answer that looks neat while quietly changing the domain of the expression.

Rational-expression fluency is also a strong preview of later work with functions and calculus. Domain restrictions, factor structure, and cancellation all reappear in more advanced contexts. This is why it is worth slowing down and writing clear restrictions now. The habit transfers directly into harder algebra and prevents later errors that are much more expensive to unwind.

Common traps

Cancelling terms that are being added or subtracted instead of cancelling factors.
Forgetting to factor fully before deciding whether cancellation is possible.
Losing denominator restrictions during simplification.
Adding fractions by combining numerators without a common denominator.

Revision focus

During revision, underline every denominator before you begin. That immediately focuses attention on possible restrictions and makes it less likely that you will cancel too aggressively. It also helps to label which step used factorisation, which step used a common denominator, and which step simplified the final result.

Another good test is to check whether the simplified expression gives the same value as the original for a simple allowed input such as $x = 1$ . If it does not, the algebra has changed the expression rather than simplifying it correctly.

Practice links

Simplification of rational algebraic expressions