Factoring Calculator

A factoring calculator rewrites a number or an algebraic expression as a product of smaller pieces. Example: 6x² + 11x + 3 has a × c = 18, and the pair 2 and 9 multiplies to 18 while adding to 11, so the trinomial factors as (3x + 1)(2x + 3). The same tool lists all 24 factors of 360 and pulls 6x² out of 12x⁵ + 18x³ − 30x².

Two sister tools cover the neighboring questions. The Prime Factorization Calculator goes deeper on whole numbers with a factor tree, so use it when you only need the prime decomposition of one integer. The Quadratic Formula Calculator solves ax² + bx + c = 0 for x, so use it when a trinomial does not factor over the integers and you still need the roots, including irrational or complex ones. Use this page when the question is "how do I write this as a product?" rather than "what is x?".

How the AC method factors a trinomial

For a trinomial ax² + bx + c with integer coefficients, the calculator looks for two integers m and n such that:

Once the pair exists, the middle term bx is rewritten as mx + nx, the four terms are grouped in pairs, the GCF is pulled out of each pair, and the shared binomial is factored out. If no such pair exists, the trinomial does not factor over the integers. The discriminant gives a quick pre-check:

Before any of this, the calculator divides out the greatest common factor of a, b and c, and factors out −1 when a is negative, so the trinomial inside the parentheses always has a positive leading coefficient and no leftover common factor.

Worked example: 6x² + 11x + 3 step by step

Here is exactly what the calculator does for its default input, with every number written out:

1. Check for a GCF. The coefficients 6, 11 and 3 share no factor greater than 1, so nothing is pulled out.
2. Multiply a × c. 6 × 3 = 18. We need two integers that multiply to 18 and add to 11.
3. Search the factor pairs of 18. 1 and 18 add to 19; 2 and 9 add to 11. Found it: m = 2, n = 9.
4. Split the middle term. 6x² + 11x + 3 = 6x² + 2x + 9x + 3.
5. Group and factor each pair. (6x² + 2x) + (9x + 3) = 2x(3x + 1) + 3(3x + 1).
6. Factor out the shared binomial. (3x + 1)(2x + 3).
7. Check by expanding. 3x · 2x = 6x², 3x · 3 = 9x, 1 · 2x = 2x, 1 · 3 = 3, and 9x + 2x = 11x. The product is 6x² + 11x + 3, so the factorization is correct.

The discriminant confirms the result in advance: D = 11² − 4 × 6 × 3 = 121 − 72 = 49, a perfect square (7²), so an integer factorization was guaranteed to exist.

Trinomials factored with the AC method

Each row below follows the same routine. The last two rows show what happens when no pair works: the discriminant is not a perfect square, and the calculator reports that the trinomial does not factor over the integers.

Notice the sign pattern: when a × c is positive, m and n share the sign of b (both positive for 6x² + 11x + 3, both negative for 2x² − 7x + 3). When a × c is negative, the two integers have opposite signs and the one with the larger absolute value carries the sign of b, as in 8x² + 2x − 15 with −10 and 12.

Worked example: all the factors of 360

Number mode uses trial division: it tests every integer d from 1 up to the square root of the number and, whenever d divides evenly, records the pair (d, n ÷ d). For 360, the square root is about 18.97, so only d = 1 through 18 need checking. Twelve of them divide 360, giving the factor pairs 1 × 360, 2 × 180, 3 × 120, 4 × 90, 5 × 72, 6 × 60, 8 × 45, 9 × 40, 10 × 36, 12 × 30, 15 × 24 and 18 × 20. Unpacking the pairs yields all 24 factors: 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180 and 360, which add up to 1,170.

The prime factorization is 360 = 2³ × 3² × 5, and it predicts the divisor count without listing anything: add 1 to each exponent and multiply, (3 + 1)(2 + 1)(1 + 1) = 4 × 3 × 2 = 24. That shortcut is why the prime factorization is the most useful single fact about a number, and it is also what the GCF Calculator and LCM Calculator rely on behind the scenes.

Factor counts and prime factorizations of common numbers

Every value in this table was produced by the same trial-division routine the calculator runs. The factor count is always even unless the number is a perfect square, where the middle pair is a number times itself.

Difference of squares: a² − b² = (a − b)(a + b)

Whenever an expression is one perfect square minus another, it factors instantly. The calculator first pulls out any numeric GCF, then checks whether both remaining parts are perfect squares. Example: 12x² − 27 has GCF 3, leaving 4x² − 9 = (2x)² − 3², so the full factorization is 3(2x − 3)(2x + 3). With a fourth power the identity applies twice: x⁴ − 81 = (x² − 9)(x² + 9) = (x − 3)(x + 3)(x² + 9). The sum of squares x² + 9 stays as it is, because a² + b² has no real factors.

GCF of a polynomial

The greatest common factor of a polynomial has two parts: the GCF of the coefficients and the lowest power of x that appears in every term. For 12x⁵ + 18x³ − 30x², the coefficients 12, 18 and 30 share the factor 6, and the smallest exponent is 2, so the GCF is 6x². Dividing each term gives 12x⁵ ÷ 6x² = 2x³, 18x³ ÷ 6x² = 3x and −30x² ÷ 6x² = −5, so the factored form is 6x²(2x³ + 3x − 5). When the leading coefficient is negative, the calculator takes the GCF as negative too: −4x⁴ + 8x² + 12x = −4x(x³ − 2x − 3). If the constant term has exponent 0, as in 7x² + 14x + 21 = 7(x² + 2x + 3), no x can be pulled out at all.

How to use this factoring calculator

1. Pick a mode. Choose Number for a whole number, ax² + bx + c for a quadratic trinomial, a² − b² for a difference of squares, or GCF for a polynomial with two to four terms.
2. Enter integers only. Type the coefficients exactly as they appear in standard form, including negative signs. For 2x² − 7x + 3 enter a = 2, b = −7, c = 3. A missing term has coefficient 0.
3. Read the headline. The blue card shows the factored form (or the prime factorization) and a verdict line saying whether the expression factors over the integers.
4. Follow the steps. The numbered list beneath the result walks through the GCF, the a × c product, the pair search, the split middle term, the grouping and the final check. Copy that reasoning into your own work.
5. Try the quick examples. The chips under each input load ready-made cases, including one that does not factor, so you can see how the calculator explains a negative result.

Who this calculator is for

• Algebra 1 and Algebra 2 students learning to factor trinomials, factor by grouping and spot special products, who want each step spelled out rather than just an answer.
• Middle-school students working with factor pairs, prime factorization and divisibility, where the Number mode lists every pair in order.
• Parents and tutors checking homework quickly or generating fresh examples with known answers.
• Test takers reviewing for the SAT, ACT, GED or a placement exam, where recognizing a difference of squares or a perfect square trinomial saves real time.
• Anyone simplifying a rational expression, since cancelling common factors starts with factoring the numerator and denominator.

Key factoring terms

• Factor: a number or expression that divides another evenly. 12 = 3 × 4, so 3 and 4 are factors of 12; (3x + 1) is a factor of 6x² + 11x + 3.
• Prime factorization: the unique way to write a whole number as a product of primes, such as 84 = 2² × 3 × 7.
• Trinomial: a polynomial with three terms. A quadratic trinomial has the form ax² + bx + c.
• Leading coefficient: the number a in front of x². When a = 1 the pair search simplifies to "two numbers that multiply to c and add to b".
• Discriminant: D = b² − 4ac. A perfect-square discriminant means the trinomial factors with whole numbers.
• Factoring by grouping: splitting a four-term expression into two pairs, factoring each pair, then factoring out the common binomial.
• Prime polynomial: a polynomial that cannot be factored over the integers, the algebraic cousin of a prime number.
• Perfect square trinomial: a trinomial of the form (px + q)², recognizable because the discriminant is 0. Example: 9x² + 12x + 4 = (3x + 2)².

What changes the result

• The sign of c: a positive c means the pair shares one sign (the sign of b); a negative c means opposite signs. That single observation halves the search.
• A common factor: 12x² + 22x + 6 and 6x² + 11x + 3 have the same binomial factors, but only the first carries the extra factor 2.
• The size of a × c: the larger the product, the more factor pairs to test. 8x² + 2x − 15 has a × c = −120 with 16 positive divisors, while x² − 7x + 12 has only 12 with 6.
• The discriminant: D = 49 factors, D = 5 does not, and D = −23 has no real roots at all. Compute it first when the pair search is not going anywhere.
• Rounding is never involved: because all arithmetic is exact integer math, the result is either correct or reported as non-factorable. There is no approximation to worry about.

Tips for factoring by hand

• Memorize the perfect squares up to 15² = 225 and the perfect cubes up to 5³ = 125; special products jump out once you know them.
• When a = 1, skip the AC step: just find two numbers that multiply to c and add to b. For x² − 7x + 12 that is −4 and −3.
• If a × c is large, list its factor pairs systematically from 1 upward rather than guessing, and stop at the square root.
• Use the sum-and-product check on your binomials: the product of the constants must equal c and the product of the x-coefficients must equal a.
• Write the GCF in front and keep it there. Losing a factor of 2 or a negative sign is the most common lost point on a test.

Limitations

• Factoring is done over the integers only. An expression such as x² − 2 factors over the reals as (x − √2)(x + √2), but the calculator reports it as non-factorable, which is the correct answer in an algebra class that works with whole numbers.
• Trinomial mode handles degree 2 in one variable. It does not factor cubics, expressions in two variables such as 6x² + 11xy + 3y², or trinomials in x⁴ disguised as quadratics.
• Coefficients must be integers between −10,000 and 10,000, and whole numbers in Number mode are limited to 1 trillion so that trial division finishes instantly.
• Sum and difference of cubes, a³ ± b³, are not implemented; those identities have their own formulas.
• The tool is a study aid. Show your own work where required and verify results by expanding.

Related calculators

To decompose one integer into primes with a factor tree, use the Prime Factorization Calculator; to solve a quadratic equation whether or not it factors, use the Quadratic Formula Calculator. The GCF Calculator and LCM Calculator handle greatest common factors and least common multiples of plain numbers, and the Polynomial Calculator multiplies polynomials, which is the fastest way to check a factorization of any degree. If you need to simplify powers before factoring, the Exponent Calculator covers the exponent rules.

Sources and method

Every result comes from standard algebraic identities and deterministic integer algorithms: trial division for factors and primes, the AC method with factoring by grouping for quadratic trinomials, the identity a² − b² = (a − b)(a + b) for differences of squares, and the coefficient-GCF plus lowest-power rule for polynomial GCFs. These are pure mathematics and need no external data; each factorization is verified by expansion before it is displayed.