Break polynomials into manageable factoring steps with explanations. See roots, checks, graphs, and exportable results. Study patterns faster with organized inputs and instant verification.
Enter the degree and coefficients. Only coefficients from the chosen degree down to the constant are used.
| Polynomial | Factored Form | Main Pattern |
|---|---|---|
| x^2 - 9 | (x - 3)(x + 3) | Difference of squares |
| x^2 + 5x + 6 | (x + 2)(x + 3) | Quadratic factoring |
| x^3 - 6x^2 + 11x - 6 | (x - 1)(x - 2)(x - 3) | Rational root theorem |
| 2x^2 - 3x + 1 | (2x - 1)(x - 1) | Rational roots |
| x^3 - 8 | (x - 2)(x^2 + 2x + 4) | Difference of cubes |
This calculator uses several factoring rules in a practical order. First, it removes the greatest common factor from all coefficients. Next, it checks whether every term contains a common variable power, such as x or x^2. That immediately simplifies many expressions.
For quadratics, it computes the discriminant, b^2 - 4ac. If the discriminant is a perfect square, the roots are rational, and the quadratic can be written as linear factors. When the middle term is zero, it also checks the difference of squares pattern.
For higher degrees, it applies the Rational Root Theorem. Possible rational roots are formed from factors of the constant term over factors of the leading coefficient. Each candidate root is tested. When a root works, synthetic-style division reduces the polynomial and the process repeats.
The calculator also recognizes simple sum of cubes and difference of cubes forms. If no valid rational root or special identity appears, the remaining factor is shown as irreducible over integers.
It works with one-variable polynomials up to degree six using coefficient inputs. It handles common factors, repeated zeros, rational roots, quadratic checks, and simple cube identities.
Yes. The calculator lists numbered steps that explain the common factor check, root testing, quotient reduction, and final factor assembly.
No. Some polynomials are irreducible over integers. In those cases, the calculator stops after valid checks and shows the remaining unfactored part.
A repeated root means the same factor appears more than once. For example, (x - 2)^2 has a double root at x = 2.
Coefficient input avoids parsing errors and makes the factoring logic more dependable. It also keeps the step list cleaner and easier to verify.
The graph shows where the polynomial crosses or touches the axis. That helps confirm real roots and gives a quick visual check.
The calculator pulls out x as a common factor. If more trailing zero coefficients exist, it extracts higher powers like x^2 or x^3.
Yes. The CSV export saves the summary and full step list. The PDF export creates a printable copy of the current factoring result.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.