Algebra · step 1 of 13

Manipulating & simplifying expressions

Simplification is the skill everything else rests on: expand brackets, collect like terms, and cancel only common FACTORS. The single most damaging habit is cancelling across a sum — writing (x + 3)/3 as x + 1 — which destroys more marks in later topics than any other error. Every cancellation must involve a factor of the whole numerator and the whole denominator.

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Method: how to approach it

The order below is what examiners expect to see, and each step carries its own marks.

Expand systematicallyMultiply each term of the first bracket by each term of the second, then collect. For three brackets, expand two first.
Collect like termsTerms match only when the variable parts are identical: 3x² and 5x² combine, 3x² and 5x do not.
Factor before cancellingCancellation is legal only between factors. Factor the numerator and denominator fully first.
Check with a test valueSubstitute x = 2 into the original and the simplified form. Different answers mean an error.

Worked example

Simplify (x² − 9)/(x² + 5x + 6).

Factor the numerator as a difference of squares: x² − 9 = (x−3)(x+3).
Factor the denominator: x² + 5x + 6 = (x+2)(x+3).
The common FACTOR (x+3) cancels.
Result: (x−3)/(x+2), valid for x ≠ −3 and x ≠ −2.

Answer. (x − 3)/(x + 2), with the restrictions x ≠ −3, −2 noted.

Where marks get dropped

These are the specific errors that cost credit on manipulating & simplifying expressions questions — QED's rubric penalises each of them separately.

Cancelling a term rather than a factor. In (x + 3)/3 the 3 is a term of the numerator, so nothing cancels.
Losing a sign when expanding a negative bracket. −(x − 4) is −x + 4, not −x − 4.
Forgetting to state excluded values after cancelling. The simplified form must record where the original was undefined.

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Manipulating & simplifying expressions — frequently asked questions

When can I cancel?

Only when the same non-zero factor multiplies the entire numerator and the entire denominator. If you cannot write both as a product containing it, you cannot cancel.

Does the domain change when I simplify?

The simplified expression may be defined where the original was not. State the excluded values so the two agree — this matters for limits and for graph sketching.

How do I check my simplification?

Substitute a convenient number that avoids the excluded values, typically x = 1 or x = 2, into both forms. Agreement is strong evidence; disagreement is proof of an error.

The rest of Algebra

Foundational algebra to close prerequisite gaps. Each subtopic below has its own method, worked example and mark-losing traps.

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