How to Factor Trinomials
Learn how to factor trinomials, find number pairs, handle leading coefficients, avoid sign mistakes, troubleshoot, and use FAQs.
Quick Answer
To factor a trinomial, look for two binomials whose first terms multiply to the first term, last terms multiply to the constant, and cross-products add to the middle term.
What You Need
- Trinomial in standard form
- Multiplication facts
- Factoring method from class
- Pencil and paper
- Way to check by expanding
Safety Precautions
- Check for a greatest common factor first.
- Keep positive and negative signs clear.
- Write the trinomial in ax squared plus bx plus c form.
- Do not skip the check step.
- Recognize when a trinomial may not factor over integers.
Step-by-Step Instructions
- Step 1
Write in standard form
Put the trinomial as ax² + bx + c.
- Step 2
Factor out GCF
Remove any greatest common factor first.
- Step 3
Handle simple leading coefficient
If a = 1, find two numbers that multiply to c and add to b.
- Step 4
Write binomials
Use those numbers in (x + m)(x + n).
- Step 5
Handle a greater than 1
Use your class method such as AC method, grouping, or trial factoring.
- Step 6
Check signs
Use the signs of b and c to guide positive and negative factors.
- Step 7
Expand to verify
Multiply the binomials to confirm the original trinomial.
Practical Example
Example: x² - 7x + 12 factors as (x - 3)(x - 4) because -3 and -4 multiply to 12 and add to -7.
Common Mistakes
- Ignoring the GCF
- Using numbers that multiply correctly but add incorrectly
- Sign errors
- Not handling a greater than 1 carefully
- Forgetting to expand-check
- Assuming every trinomial factors
Troubleshooting
You cannot find two numbers
List factor pairs of c and test their sums.
Middle term is wrong after checking
Your cross-products do not add to b; try another pair.
Leading coefficient is not 1
Use AC method or grouping instead of the simple pair method only.
No integer factors work
The trinomial may be prime over integers.
Helpful Internal Links
FAQs
What is a trinomial?
A polynomial with three terms.
What is the easiest trinomial to factor?
A trinomial where the leading coefficient is 1 is usually easiest.
How do I check my answer?
Expand the binomials and compare with the original trinomial.
What if the trinomial does not factor?
It may be prime over integers or require another method such as the quadratic formula.
Related Guides
How to Factor Polynomials
Learn how to factor polynomials by finding common factors, grouping, trinomials, special products, common mistakes, troubleshooting, and FAQs.
How to Learn Math
Learn how to learn math with foundations, examples, practice, mistake review, problem-solving habits, troubleshooting, and FAQs.
How to Calculate Volume
Learn how to calculate volume for boxes, cylinders, and simple shapes, choose the right formula, avoid unit mistakes, troubleshoot, and use FAQs.
Help us improve this guide
Tell us about an outdated instruction, unclear step, broken link or safety concern. Do not include passwords, account numbers, medical records or other sensitive information.