What the factorizer does
Enter a polynomial in any form, with brackets, fractions or several variables. The tool:
- expands it and writes it in standard form;
- takes out the greatest common factor, including powers of variables;
- recognises standard identities and repeated factors;
- finds every linear factor using the rational root theorem;
- checks whether each remaining factor can be split further;
- expands the final answer to confirm that it equals the original polynomial.
How to use it
Type the polynomial using ^ for powers, for example 3x^4 - 12x^2. Multiplication can be written as * or left implicit (3xy). Each letter is its own variable, so xy means x·y. Brackets are expanded before factoring, so (x + 1)^2 - 4 is fine.
Methods and identities
Greatest common factor
Always start by taking out the largest factor common to every term:
Standard identities
The rational root theorem
If a polynomial with integer coefficients has a rational root p/q in lowest terms, then p divides the constant term and q divides the leading coefficient. Each such root gives a factor (qx − p). For 2x² − 5x + 2, the candidates are ±1, ±2 and ±1/2. Testing shows that x = 2 and x = 1/2 are roots, so the factors are (x − 2) and (2x − 1).
Factoring by grouping
With four terms, pair them so each pair has a common factor that leaves the same bracket:
When can a factor not be split further?
A quadratic or cubic factor with no rational roots cannot be factored over the integers, because any factorisation would include a linear factor and therefore a rational root. For higher degrees, the tool tries every way of grouping the polynomial’s roots into a factor with integer coefficients, and confirms any candidate by exact division. When none works, the factor is proven irreducible.
Worked example
Factor 6x³ − 15x² + 6x completely.
- Common factor. Every term is divisible by 3x: 6x³ − 15x² + 6x = 3x(2x² − 5x + 2).
- Rational roots of the quadratic. The candidates are ±1, ±2 and ±1/2. Substituting x = 2 gives 8 − 10 + 2 = 0, and x = 1/2 gives 1/2 − 5/2 + 2 = 0.
- Write the factors. The roots 2 and 1/2 give the factors (x − 2) and (2x − 1).
- Check. 3x(2x − 1)(x − 2) = 3x(2x² − 5x + 2) = 6x³ − 15x² + 6x ✓.
Common mistakes
- Forgetting the common factor. Factor out the GCF first; otherwise the remaining factors are harder to find and the answer is incomplete.
- Stopping too early. x⁴ − 16 = (x² − 4)(x² + 4) is not finished: x² − 4 is itself a difference of squares.
- Trying to factor a sum of squares. x² + 4 does not factor over the integers. Only a difference of squares does.
- Sign errors with cubes. In a³ − b³ = (a − b)(a² + ab + b²), the sign in the quadratic is the opposite of the sign in the linear factor.
- Dropping a constant. −x² + 1 = −(x − 1)(x + 1); the leading minus sign is part of the answer.
Practical applications
- Solving equations. A factored polynomial shows its roots directly.
- Simplifying fractions. Rational expressions simplify by cancelling common factors of the numerator and denominator.
- Partial fractions. Integrating a rational function starts by factoring its denominator.
- Graph sketching. Linear factors give the x-intercepts; repeated factors show where the graph touches the axis.
Assumptions and limitations
Factorisation is over the integers, so factors such as x − √2 or x − i are not used. Polynomials in one variable, and homogeneous polynomials in two variables, are factored completely, with each factor proven irreducible. Other polynomials in several variables are factored with common factors, grouping and standard identities. When those methods find nothing further, the tool says that completeness is not guaranteed. Degrees above 40 are not supported.