Factoring Calculator

Quick Answer

Last updated: 10 November 2026 ·How we calculate this: the GCF and the rational root test

How This Factoring Calculator Works

Type a polynomial such as 6x^2 + 11x - 10 and press Calculate. The calculator puts it in standard form, takes out the greatest common factor (a number, a power of x, or both), then tests every possible rational root p/q. Each root it finds gives a linear factor, which it divides out before looking again. A quartic with only even powers is also tried as a quadratic in x². The steps table shows every stage.

How to Use the Factoring Calculator

1. Type the polynomial using x and ^ for powers.
2. Use whole-number coefficients, such as 6x^2 + 11x - 10.
3. Press Calculate.
4. Read the factored form and the roots.
5. Follow the steps table to see the GCF and each root found.

Factoring Patterns and Rules

• Difference of squares: a² − b² = (a + b)(a − b)
• Perfect square trinomial: a² ± 2ab + b² = (a ± b)²
• Sum of cubes: a³ + b³ = (a + b)(a² − ab + b²)
• Difference of cubes: a³ − b³ = (a − b)(a² + ab + b²)
• Rational root test: any rational root p/q has p dividing the constant term and q dividing the leading coefficient. A root p/q gives the factor (qx − p).

Key Terms

• Factor: a polynomial that divides another exactly.
• GCF: the greatest common factor of all the terms.
• Root (zero): a value of x that makes the polynomial equal 0.
• Prime polynomial: a polynomial that cannot be factored over the integers.
• Multiplicity: how many times a factor repeats, such as 2 in (x − 3)^2.

Worked Example

Factor 6x^2 + 11x − 10.

• GCF: the numbers 6, 11 and 10 share no common factor, so there is none to take out.
• Candidates p/q: p divides 10 (1, 2, 5, 10) and q divides 6 (1, 2, 3, 6).
• Testing gives roots x = −5/2 and x = 2/3, because 6(−5/2)² + 11(−5/2) − 10 = 0 and 6(2/3)² + 11(2/3) − 10 = 0.
• The roots give factors (2x + 5) and (3x − 2), so 6x^2 + 11x − 10 = (2x + 5)(3x − 2).
• Check by expanding: 6x^2 − 4x + 15x − 10 = 6x^2 + 11x − 10.

Real-Life Examples

• Trinomials: x^2 + 5x + 6 = (x + 3)(x + 2), because 3 × 2 = 6 and 3 + 2 = 5.
• Difference of squares: 2x^3 − 8x = 2x(x + 2)(x − 2).
• Cubes: x^3 − 27 = (x − 3)(x^2 + 3x + 9).
• Quartics: x^4 − 5x^2 + 4 = (x + 2)(x + 1)(x − 1)(x − 2), and x^4 + 5x^2 + 4 = (x^2 + 1)(x^2 + 4).

Factoring vs the Quadratic Formula

• Factoring is fastest when the roots are whole numbers or simple fractions, and gives the product form.
• The quadratic formula always works for quadratics, including irrational and complex roots, but only gives the roots.
• If the discriminant b² − 4ac is a perfect square, the quadratic factors over the integers.

Factoring Examples

Factoring Trinomials by Hand (the ac Method)

• Step 1: for ax^2 + bx + c, multiply a × c. For 6x^2 + 11x − 10, that is −60.
• Step 2: find two numbers that multiply to ac and add to b: 15 and −4.
• Step 3: split the middle term: 6x^2 + 15x − 4x − 10.
• Step 4: group: 3x(2x + 5) − 2(2x + 5).
• Step 5: take out the shared bracket: (2x + 5)(3x − 2).

Why Factor a Polynomial?

Factored form shows the roots straight away: (2x + 5)(3x − 2) = 0 when x = −5/2 or x = 2/3. That makes it the quickest way to solve many equations, to find where a graph crosses the x-axis, and to simplify fractions such as (x² − 4)/(x − 2) = x + 2.

If a quadratic will not factor over the integers, it may still have real roots. Use the quadratic formula for those; this calculator lists them as irrational or complex roots.

Common Mistakes to Avoid

• Skipping the GCF. Always take out common factors first; it makes the rest easier.
• Sign errors. The root x = 2 gives the factor (x − 2), not (x + 2).
• Thinking a² + b² factors. A sum of squares does not factor over the real numbers.
• Stopping too early. x^4 − 16 = (x^2 + 4)(x^2 − 4), and x^2 − 4 factors again.
• Not checking. Multiply the factors back out to confirm the answer.

Frequently Asked Questions

References & Sources