Calculator
Use expression mode for quick entry, or manual mode for structured inputs.
Example data table
| Binomial | Pattern | Factored form |
|---|---|---|
| x2 − 9 | Difference of squares | (x − 3)(x + 3) |
| 4x2 − 36 | GCF then squares | 4(x − 3)(x + 3) |
| 8x3 + 1 | Sum of cubes | (2x + 1)(4x2 − 2x + 1) |
| 27x3 − 64 | Difference of cubes | (3x − 4)(9x2 + 12x + 16) |
| 6x2 − 54 | GCF then squares | 6(x − 3)(x + 3) |
| −3x + 12 | GCF with sign | −3(x − 4) |
Examples are shown in a single-variable style for clarity.
Formulas used
- GCF factoring: a·m + b·m = m(a + b), where m is the greatest common factor.
- Difference of squares: A2 − B2 = (A − B)(A + B).
- Sum of cubes: A3 + B3 = (A + B)(A2 − AB + B2).
- Difference of cubes: A3 − B3 = (A − B)(A2 + AB + B2).
How to use this calculator
- Choose Expression or Manual terms.
- Select a method, or keep Auto detect.
- Enter your two-term expression and press Factor Now.
- Review the factored form, steps, and expansion check.
- Use the download buttons to export CSV or PDF.
If a special pattern does not match, the tool still applies any safe common factor.
FAQs
1) What counts as a binomial here?
A binomial has exactly two terms joined by + or −, like x² − 9 or 8x³ + 1. Multi-term expressions are outside this calculator.
2) Why does the calculator pull out a negative sign?
A leading negative can hide patterns. Factoring out −1 makes the first term positive, then common factors and identities become easier to recognize.
3) When will difference of squares work?
It works when both terms are perfect squares and the sign is subtraction, such as 4x² − 36. The tool checks square coefficients and even exponents.
4) When will sum or difference of cubes work?
It works when both terms are perfect cubes with exponents divisible by 3, such as 8x³ + 1 or 27x³ − 64. The calculator then applies the cube identities.
5) Does it fully factor every binomial?
It fully factors common classroom patterns. If your expression needs advanced number theory or multiple variables, it may only extract the safest common factor.
6) What does the expansion check mean?
The tool multiplies the factors back out as a polynomial in one variable. If the expanded result matches the original, it marks the factorization as verified.
7) Can I use variables other than x?
Yes. Use any single letter like y or t. Expression mode reads the letter from your input. Manual mode uses the variable field.
8) Why are integers recommended?
Integer coefficients make GCF, squares, and cubes detection reliable. With decimals, many classic factoring identities still exist, but matching becomes less predictable.