How to check trinomial factorisation
Use this page when you have already factored a trinomial and want a fast way to confirm the brackets are actually correct.
Start here if you want the short version before reading the full method.
- The quickest check is to expand the brackets and compare the result term by term with the original trinomial.
- You should match the x^2 term, the middle x term, and the constant term exactly.
What this topic means and what to look for first.
Many factorisation errors look nearly right because the first and last terms match while the middle term is off by a sign or a number.
A good check does more than glance at the answer. It rebuilds the trinomial and compares each part carefully.
One reliable route through the topic.
- 1Write the original trinomial and the claimed factorisation side by side.
- 2Expand the brackets fully instead of checking mentally.
- 3Combine like terms to rebuild a single trinomial.
- 4Compare the x^2 term, the middle x term, and the constant term with the original.
- 5If one part does not match, trace whether the issue comes from the factor pair or the sign pattern.
Choose the route that fits the quadratic.
Best for almost every trinomial because it shows exactly where a mismatch starts.
Useful when a = 1 and you want to confirm the chosen pair multiplies and adds correctly before expanding.
See the method in action.
Check whether (x + 2)(x + 3) is the correct factorisation of x^2 + 5x + 6.
- Expand the brackets: (x + 2)(x + 3) = x^2 + 3x + 2x + 6.
- Combine the middle terms to get x^2 + 5x + 6.
- Because every term matches, the factorisation is correct.
Check whether (x + 1)(x + 6) is the correct factorisation of x^2 + 5x + 6.
- Expand the brackets: (x + 1)(x + 6) = x^2 + 6x + x + 6.
- Combine the middle terms to get x^2 + 7x + 6.
- The constant matches but the middle term does not, so this factorisation is not correct.
Check whether (x - 4)(x - 3) is the correct factorisation of x^2 - x - 12.
- Expand the brackets: (x - 4)(x - 3) = x^2 - 3x - 4x + 12.
- Combine the middle terms to get x^2 - 7x + 12.
- Both the middle term and the constant fail to match, so the sign pattern is wrong.
Things that commonly send the method off track.
- Checking only the constant term and forgetting that the middle term must match too.
- Expanding mentally and missing a sign change.
- Stopping when the factor pair looks plausible instead of rebuilding the full trinomial.
Use a short verification pass before moving on.
- Expand each bracket term carefully and write every intermediate line.
- If the rebuilt middle term is wrong, check both the number pair and the bracket signs.
- If the x^2 term changes unexpectedly, the bracket starts were set up incorrectly.
Try a few variations before switching to a calculator or solver tool.
- Check whether (x + 4)(x + 2) matches x^2 + 6x + 8.
- Check whether (x - 5)(x + 2) matches x^2 - 3x - 10.
- Check whether (2x + 1)(x + 3) matches 2x^2 + 7x + 3.
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Short answers worth checking.
Expand the brackets, combine like terms, and compare the rebuilt trinomial with the original expression term by term.
That usually means the factor pair or the sign pattern is wrong, even though the brackets may look close at first glance.
Continue with the next closely related topic.
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