Enter Trinomial Coefficients
Use integer coefficients for reliable algebraic factoring over the integers. The graph still appears for every valid quadratic.
Example Data Table
| Input Trinomial | Factored Form | Notes |
|---|---|---|
| x^2 - 5x + 6 | (x - 2)(x - 3) | Simple monic example with two positive roots. |
| 2x^2 + 7x + 3 | (2x + 1)(x + 3) | Non-monic example using the ac method. |
| 3x^2 - 12x + 12 | 3(x - 2)(x - 2) | Includes a common factor and repeated root. |
| x^2 + x + 1 | Prime over integers | No integer pair matches the product-sum condition. |
Use the examples to compare monic, non-monic, repeated-root, and prime cases before entering your own coefficients.
Formula Used
This calculator uses the standard quadratic form and then applies algebraic factoring rules in a strict order.
If no integer pair meets the product-sum condition after removing the greatest common factor, the trinomial is reported as prime over the integers.
How to Use This Calculator
- Enter integer values for coefficients a, b, and c.
- Choose the variable symbol you want displayed.
- Set the graph window with minimum and maximum x values.
- Pick graph points and decimal precision for the output.
- Press Factor Trinomial to place the result under the header.
- Read the fully factored form, roots, discriminant, and vertex.
- Review the factoring steps to understand the algebra process.
- Use the CSV and PDF buttons to export the calculated result.
Frequently Asked Questions
1. What kind of expressions can this calculator factor?
It handles quadratic trinomials in standard form, ax² + bx + c, with integer coefficients. It first removes any greatest common factor and then tests whether the reduced trinomial factors over the integers.
2. Why does the calculator say prime over the integers?
That message appears when no integer pair satisfies the product-sum condition after any common factor is removed. The quadratic may still have irrational or complex roots, but it does not split into integer binomials.
3. Why is the greatest common factor removed first?
Removing the greatest common factor simplifies the coefficients and exposes the real factoring structure. This is standard algebra practice because many trinomials only reveal their simplest factor pattern after the common factor is taken out.
4. What does the discriminant tell me?
The discriminant, b² - 4ac, tells you how many real roots exist. Positive means two real roots, zero means one repeated root, and negative means a complex conjugate pair.
5. Can I use this for non-monic trinomials?
Yes. Non-monic cases, where a is not 1, are supported. The calculator applies the ac method and searches for integer binomial factors that multiply back to the original leading coefficient and constant.
6. Why is a graph included in a factoring calculator?
The graph shows where the quadratic crosses the x-axis, where the vertex lies, and how the signs of the factors behave. It makes the algebra result easier to verify visually.
7. Does the variable symbol change the math?
No. Changing the variable only changes the displayed letter. The factoring rules, root calculations, discriminant, and graph remain exactly the same because the coefficients determine the algebraic behavior.
8. What exports are available after calculation?
You can export the result table as a CSV file and capture the result section as a PDF. Both buttons appear after a valid calculation is completed.