Enter a cubic polynomial
Formula used
A general cubic polynomial has the form:
The calculator first checks common factors and standard identities.
a³ − b³ = (a − b)(a² + ab + b²)
a³ + b³ = (a + b)(a² − ab + b²)
For general cubics, rational candidates follow ±p/q. Here, p divides the constant and q divides the leading coefficient.
How to use this calculator
- Select expression mode or separate coefficient mode.
- Enter a valid cubic polynomial using one variable.
- Choose the desired number system and output style.
- Select the steps, candidates, roots, and verification details.
- Press “Factor expression” to calculate the complete result.
- Copy, print, or export the displayed solution.
Example cubic expressions
| Expression | Factored form | Primary method |
|---|---|---|
| x³ − 6x² + 11x − 6 | (x − 1)(x − 2)(x − 3) | Rational roots |
| x³ − 8 | (x − 2)(x² + 2x + 4) | Difference of cubes |
| x³ + 27 | (x + 3)(x² − 3x + 9) | Sum of cubes |
| x³ + 3x² + 3x + 1 | (x + 1)³ | Repeated root |
| 2x³ − 8x | 2x(x − 2)(x + 2) | Common factor |
Understanding cubic factorization
Cubic expressions contain a highest exponent of three. Their graphs may cross the horizontal axis once or three times. Repeated roots can make the curve touch before turning.
Factoring rewrites one polynomial as simpler multiplying parts. Exact factors reveal roots without repeated numerical testing. They also make graph behavior much easier to interpret.
The first check should always search for common factors. Removing a common number or variable simplifies every later step. Special cube identities can then produce immediate answers.
General cubics often need the Rational Root Theorem. It creates a finite list of possible rational roots. Each candidate is tested by substitution or synthetic division.
A zero remainder confirms a linear factor. Synthetic division then reduces the cubic into a quadratic. The remaining quadratic may factor or require its formula.
Not every cubic has three rational roots. Some contain irrational or complex solutions. Selecting the real or complex domain displays numerical linear factors.
The discriminant helps classify the roots. A positive value indicates three distinct real roots. A negative value indicates one real and two complex roots.
A zero discriminant signals repeated roots. Multiplicity describes how many times one root appears. Odd multiplicities cross, while even multiplicities usually touch.
Exact fractions preserve algebraic accuracy during rational calculations. Decimal output is useful for graphing and estimation. Increasing precision shows more digits but not more certainty.
Always verify a factorization by expanding every factor. The expanded coefficients must match the original expression exactly. This calculator performs that comparison for dependable study results.
Frequently asked questions
What is a cubic expression?
A cubic expression is a polynomial whose highest variable power is three.
Can this calculator factor decimals and fractions?
Yes. It converts valid decimal and fractional coefficients into exact rational values.
What happens when no rational root exists?
The calculator leaves an irreducible factor or reports numerical real and complex roots.
Why are complex roots displayed?
Every cubic has three complex roots when multiplicity is counted.
What does root multiplicity mean?
Multiplicity shows how many times the same linear factor appears.
How does synthetic division help?
It quickly divides a polynomial by a tested linear factor.
What is the Rational Root Theorem?
It lists possible rational roots using factors of two coefficients.
Can I use another variable besides x?
Yes. Enter one alphabetic variable such as y, t, or z.
How can I verify the answer?
Expand the reported factors and compare every resulting coefficient.