Calculator Inputs
Example Data Table
| Method | Example expression | Factored result | Observation |
|---|---|---|---|
| Quadratic | x2 + 5x + 6 | (x + 2)(x + 3) | Middle term comes from cross products. |
| Common factor | 6x4 + 9x3 + 15x2 | 3x2(2x2 + 3x + 5) | Coefficient GCF and minimum exponent both matter. |
| Difference of squares | 9x2 − 16 | (3x − 4)(3x + 4) | Use u2 − v2. |
| Difference of cubes | 8x3 − 27 | (2x − 3)(4x2 + 6x + 9) | The second bracket changes signs by formula. |
| Cubic | x3 − 6x2 + 11x − 6 | (x − 1)(x − 2)(x − 3) | Rational root testing finds exact linear factors. |
Formula Used
Greatest common factor: For terms sharing a coefficient divisor and the same variable, factor out gcd(coefficients) and the smallest shared exponent.
Quadratic trinomial: Factor ax2 + bx + c by matching products and sums. The discriminant is b2 − 4ac.
Difference of squares: u2 − v2 = (u − v)(u + v).
Sum of cubes: u3 + v3 = (u + v)(u2 − uv + v2).
Difference of cubes: u3 − v3 = (u − v)(u2 + uv + v2).
Cubic polynomial: Rational root candidates come from factors of the constant term over factors of the leading coefficient. After one root is found, synthetic division reduces the degree.
How to Use This Calculator
Choose the factoring method that matches your expression pattern. Enter one variable letter and fill the visible inputs only.
Click Factor Expression. The result appears below the header and above the form with the original expression, factored form, steps, and graph.
Use the CSV button to save a compact worksheet record. Use the PDF button to save a printable summary for notes, revision, or homework support.
FAQs
1. What types of expressions can this page factor?
It handles greatest common factors, quadratic trinomials, difference of squares, sum or difference of cubes, and many cubic polynomials using rational-root testing.
2. Why does the calculator sometimes show decimal factors?
Some expressions do not split into integer factors. In those cases, the calculator shows real-number factors or approximate roots when a real factorization still exists.
3. What does the discriminant tell me?
For quadratics, the discriminant tells whether the roots are distinct, repeated, or non-real. It also hints whether factoring over the real numbers is possible.
4. Can I use variables other than x?
Yes. Enter any single letter variable such as x, y, t, or z. The displayed expression and factors update to match your chosen symbol.
5. Does the graph represent the factored form or the original expression?
The graph represents the original polynomial expression. Because equivalent forms produce the same curve, the plot still confirms the factorization visually through intercept behavior.
6. Why is a cubic not always fully split into three exact factors?
Some cubics have irrational or complex roots. When no rational factor exists, the tool shows approximate real roots or a real linear factor with a remaining quadratic.
7. What should I enter for the common factor method?
Enter the coefficient and exponent for each nonzero term. The tool then finds the greatest shared coefficient divisor and smallest shared exponent automatically.
8. Can I download my work for class or revision?
Yes. The CSV export saves a compact data record, while the PDF export creates a neat summary containing the expression, factors, and method steps.