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Polynomial Factoring Calculator

Break a polynomial into its simplest factors and find its roots exactly, using common factors, the rational root test and synthetic division, with every step shown.

Level: Grades 10–11

  • Common Core standard HSA.SSE.A.2Use the structure of an expression to identify ways to rewrite it.
  • Common Core standard HSA.APR.B.2Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x - a is p(a), so p(a) = 0 if and only if (x - a) is a factor of p(x).
  • Common Core standard HSA.APR.B.3Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.
Printable worksheet

Polynomial Factoring Calculator

Inputs

Use ^ for powers, for example x^4 - 5x^2 + 4. Products and brackets such as (x - 1)(x^2 + 2) are expanded first. Up to degree 12.

Result

Factored form

(2x − 1)(x + 2)(x − 3)

2x^3 − 3x^2 − 11x + 6 factors as (2x − 1)(x + 2)(x − 3).

Rational roots
x = 1/2, x = −2, x = 3
Degree
3
Status
Completely factored over the rationals
Expanded form
2x^3 − 3x^2 − 11x + 6

See the step-by-step solution ↓

Step-by-step solution

  1. Write the polynomial in standard form, highest power first

    p(x) = 2x^3 − 3x^2 − 11x + 6

  2. By the rational root theorem, any rational root is ±(a factor of 6) ÷ (a factor of 2)

    1/2, −1/2, 1, −1, 3/2, −3/2, 2, −2, 3, −3, 6, −6

  3. Test x = 1/2: p(1/2) = 0, so (2x − 1) is a factor. Dividing it out (synthetic division) leaves x^2 − x − 6

    2x^3 − 3x^2 − 11x + 6 = (2x − 1)(x^2 − x − 6)

  4. Test x = −2: p(−2) = 0, so (x + 2) is a factor. Dividing it out (synthetic division) leaves x − 3

    x^2 − x − 6 = (x + 2)(x − 3)

  5. Put the factors together. Multiplying them back out gives the original polynomial

    2x^3 − 3x^2 − 11x + 6 = (2x − 1)(x + 2)(x − 3)

Test yourself

20 quick questions built from this calculator's formula: work out results, predict what happens when an input changes, and work backward from an answer. Every answer comes with the working.

What this calculator does

Type in a polynomial and get it broken down into its simplest pieces, along with every value of x that makes it zero. A polynomial is an expression built by adding up powers of x, each times a number. Factoring means rewriting it as smaller pieces multiplied together, the way 12 can be written as 2 × 2 × 3. The factored form shows the roots (the x values that make it zero) at a glance. If p(x) = (2x − 1)(x + 2)(x − 3), then p(x) is zero exactly when x is 1/2, −2 or 3. That's why factoring is the main tool for solving polynomial equations, simplifying fractions of polynomials and sketching graphs.

You can type any polynomial in x, up to degree 12 (the highest power of x). The calculator factors it exactly over the rationals, meaning it only uses whole numbers and fractions, with no rounding. First it pulls out the greatest common factor and any power of x. Then it lists the possible fraction roots that the rational root theorem allows, and tests each one. Every root it finds is divided out by synthetic division (a quick shortcut for dividing), and repeated roots are caught too. Quartics (degree 4) with no fraction roots are checked for a split into two quadratics. Finally it gives the roots of whatever is left. Quadratics get exact answers with square roots. Anything larger gets clearly marked approximations.

Where it's used

Why this math matters outside the classroom:

  • Solving equations. An algebra student solves 2x³ − 3x² − 11x + 6 = 0 by factoring it as (2x − 1)(x + 2)(x − 3) to get x = 1/2, −2 and 3.
  • Graph sketching. Students sketch y = x⁴ − 5x² + 4 by factoring it into (x − 1)(x + 1)(x − 2)(x + 2), so the graph crosses the axis at ±1 and ±2.
  • Box volume problems. A designer finds the cut size for an open box of volume 1,000 cm³ by factoring the cubic volume equation.
  • Simplifying fractions. A student simplifies (x² − 9) ÷ (x² + 5x + 6) by factoring both to (x − 3)(x + 3) and (x + 2)(x + 3), then cancelling.
  • Control engineering. Engineers factor a system's characteristic polynomial to find its roots, which show whether the system settles or oscillates.

Grade level and standards: Grades 10–11; Common Core State Standards for Mathematics HSA.SSE.A.2, HSA.APR.B.2, HSA.APR.B.3. © Copyright 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved.

How to use it

  1. Type the polynomial using x as the variable and ^ for powers, for example 2x^3 - 3x^2 - 11x + 6. Products and brackets are fine; the calculator multiplies them out first.
  2. Read the factored form at the top. Below it are the rational roots (whole numbers and fractions) and any irrational or complex roots.
  3. Check the status line. 'Completely factored' means no piece can be broken down further using whole numbers and fractions.
  4. Open the working to follow along: the common factor, the list of possible roots, each test that worked and what's left after each division.

What the inputs mean

Polynomial in x
Fractions and decimals are turned into exact fractions, so 0.5x^2 - 0.5 factors as (1/2)(x − 1)(x + 1). To multiply, write things side by side or use *, as in 3x(x + 1).

Formula and method

Rational root theorem: if p/q (in lowest terms) is a root of aₙxⁿ + ... + a₀, then p divides a₀ and q divides aₙ
Factor theorem: p(r) = 0  ⇔  (x − r) is a factor of p(x);  for r = p/q the integer factor is (qx − p)
Synthetic division: p(x) = (x − r) × q(x), with q(x) one degree lower
Leftover quadratic ax² + bx + c: roots x = (−b ± √(b² − 4ac)) ÷ 2a
Quartic split: ax⁴ + ... + e = (a₁x² + b₁x + c₁)(a₂x² + b₂x + c₂), with a₁a₂ = a and c₁c₂ = e

Here's what the letters mean. aₙ is the leading coefficient, the number in front of the highest power xⁿ, and a₀ is the constant term, the plain number at the end. p/q is a possible fraction root, with p on top and q on the bottom. r is any root, and q(x) is what's left after dividing. In the last two lines, a, b, c and e are the polynomial's own numbers, and a₁, b₁, c₁, a₂, b₂, c₂ are the numbers in the two quadratic pieces.

The first step is always the greatest common factor. The calculator divides every coefficient (the number in front of each term) by the largest number they share. If every term contains x, it takes out x as well. That leaves a simpler polynomial with whole-number coefficients.

The rational root theorem turns the hunt for fraction roots into a short list. Each candidate is ± a factor of the constant term divided by a factor of the leading coefficient. The calculator tests each candidate by plugging it in exactly. When one gives zero, the factor theorem says a matching first-degree piece divides the polynomial, and synthetic division removes it. The same root is tried again, in case it appears more than once.

Once all fraction roots are gone, what's left has no fraction roots. A quadratic or cubic in that state can't be factored with whole numbers and fractions, so it's called irreducible. A quartic might still be two quadratics multiplied together, so the calculator checks every possible pair of whole-number quadratics.

Leftover pieces still have real or complex roots. For a quadratic, the quadratic formula gives them exactly. For higher degrees they're found by number-crunching and marked with ≈.

Worked example

Factor p(x) = 2x³ − 3x² − 11x + 6.

  1. List the candidates: ± a factor of 6 over a factor of 2. That gives 1/2, 1, 3/2, 2, 3 and 6, each with either sign.
  2. Test x = 1/2: 2(1/8) − 3(1/4) − 11(1/2) + 6 = 1/4 − 3/4 − 11/2 + 6 = 0. It works, so (2x − 1) is a factor.
  3. Divide it out: 2x³ − 3x² − 11x + 6 = (2x − 1)(x² − x − 6).
  4. Test x = −2 on what's left: 4 + 2 − 6 = 0, so (x + 2) is a factor. That leaves x² − x − 6 = (x + 2)(x − 3).
  5. Multiply back to check: (2x − 1)(x + 2)(x − 3) = (2x − 1)(x² − x − 6) = 2x³ − 3x² − 11x + 6.

Result: 2x³ − 3x² − 11x + 6 = (2x − 1)(x + 2)(x − 3), with roots x = 1/2, x = −2 and x = 3.

Understanding the result

Each first-degree factor (qx − p) gives a root x = p/q. A factor raised to a power gives a repeated root. For an even power, the graph touches the x-axis there instead of crossing it.

An irreducible quadratic factor such as x² + 2x + 4 has either no real roots or two irrational ones. You can't split it using fractions. You can split it using square roots or complex numbers, and the roots list shows those.

'Completely factored over the rationals' is what algebra courses mean by 'factor completely'. If you allow all real numbers, x² − 2 would split further as (x − √2)(x + √2).

Assumptions and limitations

  • The polynomial has one variable, x, and rational coefficients (whole numbers, fractions or decimals that end).
  • Factoring is over the rationals: every factor has fraction coefficients, which can always be scaled up to whole numbers.
  • Degree 12 at most. For degree 5 and above, pieces with no fraction roots aren't searched further, so the status line says the result may be partial.
  • Very large coefficients (beyond about a trillion) stop the search for fraction roots.
  • Number-crunched roots of high-degree leftovers are approximations. They're usually accurate to many decimal places, but not exact.
  • Polynomials in more than one variable, such as x² − y², aren't supported.

Frequently asked questions

How do you factor a trinomial like x² + 5x + 6?

Find two numbers that multiply to the constant 6 and add to the middle number 5. Those are 2 and 3, so x² + 5x + 6 = (x + 2)(x + 3). When the number in front of x² isn't 1, as in 6x² + x − 2, the rational root theorem finds the roots 1/2 and −2/3. That gives (2x − 1)(3x + 2).

What is the rational root theorem?

It narrows down which fractions could be roots. Take a polynomial with whole-number coefficients and a fraction root p/q in lowest terms. Then p divides the constant term and q divides the leading coefficient. For 2x³ − 3x² − 11x + 6, that limits the possible fraction roots to ±1, ±2, ±3, ±6, ±1/2 and ±3/2.

What does it mean if a polynomial can't be factored?

It's irreducible over the rationals: no product of polynomials with fraction coefficients equals it. x² + 1 and x² − 2 are examples. They still have roots, but those roots are complex (±i) or irrational (±√2).

How do I factor a perfect square trinomial?

A perfect square trinomial looks like a² + 2ab + b² or a² − 2ab + b². It factors as (a + b)² or (a − b)². For example, x² + 10x + 25 = (x + 5)², with the repeated root x = −5. The calculator shows the square and how many times the root repeats.

Can this factor the sum or difference of cubes?

Yes. x³ − 8 has the root 2, so it factors as (x − 2)(x² + 2x + 4). That matches the difference-of-cubes pattern a³ − b³ = (a − b)(a² + ab + b²). The quadratic piece's roots are the complex numbers −1 ± i√3.

How this calculator is checked

Method: Factoring over the rationals by the greatest common factor, the rational root and factor theorems with exact synthetic division (BigInt rational arithmetic), an exact search for integer quadratic pairs of quartics (Gauss's lemma), the quadratic formula, and Durand–Kerner iteration for numerical roots. Every result is multiplied back out and checked against the input. The calculation is covered by automated tests with known reference values and edge cases. Last updated 2026-10-18. Found a problem? Tell us.

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