Factoring Calculator
Enter a polynomial and get its complete factorization over the integers, with each step written out: the greatest common factor, then difference of squares, perfect squares, sums and differences of cubes, the AC method for ax² + bx + c, grouping, or the rational root theorem. The factors are multiplied back out as a check, and a polynomial that cannot be factored is called prime. Free, no sign-up.
One letter, for example 6x^2 + x − 12, x^3 + 8 or 2x^3 − 3x^2 − 11x + 6. Powers use ^ or superscripts.
Factored form
(3x − 4)(2x + 3)
Factorization
Factored completely
Status
Step by step
AC method: multiply a × c = 6 × (−12) = −72. Find two numbers that multiply to −72 and add to b = 1: they are −8 and 9.
Split the middle term using those two numbers.
6x² + x − 12 = 6x² − 8x + 9x − 12
Group the terms in pairs and take the common factor out of each pair.
2x(3x − 4) + 3(3x − 4)
Both pairs now share a bracket; factor it out.
(3x − 4)(2x + 3)
Check by expanding
Multiplying the factors back out gives 6x² + x − 12, the polynomial you entered.
How to Use the Factoring Calculator
- 1. Type the polynomial. For example 6x^2 + x − 12 or x^3 + 8. Use one letter; powers can be written with ^ or as superscripts, and brackets are expanded first.
- 2. Read the factored form. It appears as a product, with repeated factors written as powers, and the calculator says whether the polynomial is prime.
- 3. Follow the steps. Each line names the method used and shows the polynomial before and after, in the order a teacher would try them.
- 4. Check. The factors are expanded and compared with what you typed.
The Order to Try Methods In
| Look for | Pattern | Factors as |
|---|---|---|
| A common factor in every term | ab + ac | a(b + c) — always first, including a minus sign if the leading term is negative |
| Two terms, both perfect squares, subtracted | A² − B² | (A − B)(A + B) |
| Two terms, both perfect cubes | A³ + B³ or A³ − B³ | (A + B)(A² − AB + B²) or (A − B)(A² + AB + B²) |
| Three terms, first and last perfect squares | A² ± 2AB + B² | (A ± B)² |
| Three terms, ax² + bx + c | Two numbers with product ac and sum b | The AC method, below |
| Four terms | Pairs share a bracket | Factor by grouping |
| Anything else | A rational root p/q | Divide out (qx − p) and repeat |
The AC Method: 6x² + x − 12
- Multiply a and c. 6 × (−12) = −72.
- Find two numbers that multiply to −72 and add to b = 1: they are −8 and 9.
- Split the middle term. 6x² + x − 12 = 6x² − 8x + 9x − 12.
- Group and factor each pair. 2x(3x − 4) + 3(3x − 4).
- Factor out the shared bracket. (3x − 4)(2x + 3).
When a = 1 the method collapses to the familiar shortcut: for x² + 5x + 6 find two numbers with product 6 and sum 5 — 2 and 3 — and write (x + 2)(x + 3). If no such integer pair exists, the discriminant b² − 4ac is not a perfect square and the quadratic is prime over the integers.
Other Worked Examples
| Polynomial | Method | Factored form |
|---|---|---|
| 12x³ − 27x | GCF 3x, then difference of squares (2x)² − 3² | 3x(2x − 3)(2x + 3) |
| 27x³ − 1 | Difference of cubes (3x)³ − 1³ | (3x − 1)(9x² + 3x + 1) |
| x³ + 2x² + 3x + 6 | Grouping: x²(x + 2) + 3(x + 2) | (x + 2)(x² + 3) |
| 2x³ − 3x² − 11x + 6 | Rational root x = 1/2, then the quadratic | (x − 3)(2x − 1)(x + 2) |
| x⁴ + 4 | Two quadratics (Sophie Germain's identity) | (x² − 2x + 2)(x² + 2x + 2) |
What “Prime” Means Here
A polynomial is prime (or irreducible) over the integers when it cannot be written as a product of lower-degree polynomials with whole-number coefficients. x² + 4 and x² + x + 1 are prime. Prime is relative to the number system: x² − 2 is prime over the integers but factors as (x − √2)(x + √2) over the real numbers, and x² + 4 factors as (x − 2i)(x + 2i) over the complex numbers. This calculator factors over the rationals, which for whole-number polynomials gives the same result as factoring over the integers.
Limits
Quadratics and cubics are always factored completely. For degree 4 and above the calculator uses patterns, grouping, rational roots and a search for two quadratic factors. A factor that none of these splits is reported as left over, with a note, rather than called prime.
Common Mistakes
- Skipping the GCF. 2x² + 10x + 12 is easier as 2(x² + 5x + 6) = 2(x + 2)(x + 3). Always take the common factor out first.
- Sign errors in the AC step. If c is negative the two numbers have opposite signs; if c is positive they share a sign — the sign of b.
- Trying to factor a sum of squares. x² + 9 is prime over the integers; only a difference of squares factors.
- Mixing up the cube formulas. The middle sign of the second bracket is always the opposite of the sign in the first: A³ + B³ = (A + B)(A² − AB + B²) and A³ − B³ = (A − B)(A² + AB + B²).
- Stopping early. x⁴ − 16 = (x² − 4)(x² + 4) is not finished: x² − 4 factors again.
Frequently Asked Questions
- How do you factor a trinomial ax² + bx + c?
- Use the AC method. Multiply a and c, find two numbers that multiply to ac and add to b, split the middle term with them, factor each pair, and take out the shared bracket. When a = 1 you only need two numbers with product c and sum b.
- What is the difference of squares formula?
- A² − B² = (A − B)(A + B). It applies when a binomial is a perfect square minus a perfect square, such as 9x² − 25 = (3x − 5)(3x + 5). A sum of squares such as x² + 25 does not factor over the integers.
- What are the sum and difference of cubes formulas?
- A³ + B³ = (A + B)(A² − AB + B²) and A³ − B³ = (A − B)(A² + AB + B²). For 27x³ − 1, A = 3x and B = 1, so it factors as (3x − 1)(9x² + 3x + 1).
- How do I know if a polynomial is prime?
- For a quadratic, check the discriminant b² − 4ac: if it is not a perfect square, there are no rational roots and the quadratic is prime over the integers. A cubic is prime when it has no rational root. For higher degrees, no simple test exists.
- What is factoring by grouping?
- For four terms, split them into two pairs, take the greatest common factor out of each, and if the brackets match, factor the bracket out. x³ + 2x² + 3x + 6 becomes x²(x + 2) + 3(x + 2) = (x + 2)(x² + 3).
- Why is the common factor negative sometimes?
- When the leading coefficient is negative the calculator takes out a negative common factor, so the polynomial inside the bracket starts positive. −2x² + 8 becomes −2(x² − 4) = −2(x − 2)(x + 2). Either convention is correct; check which your course expects.
- Can I check a factorization myself?
- Yes, multiply the factors back out. If you get the original polynomial, the factorization is right. The calculator does this for you in the final check.