Factoring trinomials with steps
Use this page when you want the factoring process itself, not just the final brackets.
Start here if you want the short version before reading the full method.
- Find the number pattern first, then build the brackets and expand them again to check the result.
- A valid factorisation must satisfy both the multiplication condition and the middle-term condition.
What this topic means and what to look for first.
This page is the practical worked route for users who searched specifically for the steps.
The goal is to show the method cleanly enough that you can reuse it on the next trinomial without guessing.
One reliable route through the topic.
- 1List the useful factor pairs for the constant term, or for ac if the first coefficient is not 1.
- 2Choose the pair that also rebuilds the middle term correctly.
- 3Write the bracket form and check the sign pattern carefully.
- 4Expand the brackets again to make sure the original trinomial returns exactly.
See the method in action.
x^2 + 8x + 15
- The factor pairs of 15 are 1 and 15, and 3 and 5.
- Only 3 and 5 add to 8, so the factorisation is (x + 3)(x + 5).
- Expand to check: x^2 + 5x + 3x + 15 = x^2 + 8x + 15.
x^2 - 7x + 12
- Look for two positive numbers that multiply to 12 and add to -7 after the signs are included.
- The pair is -3 and -4, so the factorisation is (x - 3)(x - 4).
- The negative middle term tells you both bracket signs must be negative here.
Things that commonly send the method off track.
- Stopping after the product check and forgetting to test the middle term.
- Using the right numbers with the wrong sign pattern.
- Skipping the expansion check and trusting the brackets too early.
Use a short verification pass before moving on.
- Expand the brackets term by term and combine like terms.
- Compare the rebuilt expression to the original trinomial, especially the sign of the middle term.
Try a few variations before switching to a calculator or solver tool.
- x^2 + 9x + 20
- x^2 - 5x - 14
- x^2 - 11x + 24
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Extra algebra revision resources
If you want more printed algebra practice after this page, these broader searches are a sensible next step.
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Algebra workbook and revision book search
Useful if you want more equation, factorising, and worked-example practice in one printed source.
View Algebra workbook and revision book searchAmazon
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A wider GCSE-style search if you want more mixed algebra questions beyond one online guide.
View GCSE algebra practice resources searchNeed live help with factorising trinomials?
If the sign logic or bracket choices still do not feel stable, send a short enquiry describing the exact kind of trinomial that is causing trouble.
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Short answers worth checking.
The product alone is not enough because many number pairs multiply correctly but produce the wrong middle term.
Yes. If the constant term is positive and the middle term is negative, both bracket signs are often negative.
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