Factoring Calculator

Result
\left(3x + 2\right)\left(2x - 1\right)

The factored form is \displaystyle \left(3x + 2\right)\left(2x - 1\right).

  • Type: Polynomial
  • Method: Factor
  • Factors: \displaystyle \left(3x + 2\right), \displaystyle \left(2x - 1\right)
  • Zeros: \displaystyle -\frac{2}{3},\ \frac{1}{2} (where it equals 0)

Step-by-step solution

The problem

6x^{2} + x - 2
1

Use the AC method

\begin{aligned} -3 \times 4 &= -12 \\[4pt] -3 + 4 &= 1 \end{aligned}

Find two numbers that multiply to \displaystyle a \cdot c = 6 \cdot \left(-2\right) = -12 and add to \displaystyle b = 1: they are \displaystyle -3 and \displaystyle 4.

2

Split the middle term

6x^{2} + x - 2 = 6x^{2} - 3x + 4x - 2

Write \displaystyle x as \displaystyle -3x + 4x.

3

Factor by grouping

\begin{aligned} &= \left(6x^{2} - 3x\right) + \left(4x - 2\right) \\[4pt] &= 3x\left(2x - 1\right) + 2\left(2x - 1\right) \\[4pt] &= \left(3x + 2\right)\left(2x - 1\right) \end{aligned}

Take the common factor out of the first two terms and out of the last two – the same bracket is left in both.

4

Fully factored

6x^{2} + x - 2 = \left(3x + 2\right)\left(2x - 1\right)

Graph

  • \displaystyle y = 6x^{2} + x - 2

The graph of \displaystyle y = 6x^{2} + x - 2. Its zeros, turning points and any holes are marked.

Crosses the x-axis at
x = −2/3, 1/2
Crosses the y-axis at
(0, −2)
Lowest point (vertex)
(−0.083, −2.042)
Axis of symmetry
x = −1/12
Opens
upwards (a U shape)
−2−112−101020

Type a polynomial like 6x^2 + x - 2 or x^3 - 8. Use ^ for powers.

Try an example

Trinomials x^2 - 5x + 66x^2 + x - 2x^2 + 5xy + 6y^2
Special 4x^2 - 9x^3 - 8x^2 + 6x + 9
Grouping x^3 + x^2 - 4x - 4x^2 + 6x + 9 - y^2
Higher degree 2x^3 - 3x^2 - 11x + 6x^4 - 16x^6 - 64

Type a polynomial and it is factored the way it is taught – common factor first, then the pattern that fits: 6x² + x − 2 = (2x − 1)(3x + 2) by the AC method. It also spots differences of squares, perfect squares and sums or differences of cubes, groups four-term polynomials and uses the rational root theorem on cubics.

The factoring checklist

Work through these in order – it is the order the calculator uses:

  1. Greatest common factor. Always first: 12x^2y - 18xy^2 = 6xy(2x - 3y).
  2. Two terms. Difference of squares a^2 - b^2 = (a - b)(a + b); sum or difference of cubes a^3 \pm b^3 = (a \pm b)(a^2 \mp ab + b^2). A sum of squares a^2 + b^2 does not factor.
  3. Three terms. A perfect square a^2 \pm 2ab + b^2 = (a \pm b)^2? Otherwise find two numbers that multiply to ac and add to b – directly when a = 1, by splitting the middle term and grouping (the AC method) when a ≠ 1. The same works for x^4 + bx^2 + c, treating x^2 as the variable.
  4. Four terms. Group them in pairs; or spot a perfect square trinomial minus a square.
  5. Higher degree. Use the rational-root theorem to find a root r, divide by (x - r) with synthetic division, and factor what is left.
  6. Check every factor again – x^4 - 16 = (x^2 - 4)(x^2 + 4) still has x^2 - 4 = (x - 2)(x + 2) inside.

Special patterns at a glance

PatternFactoredExample
Difference of squaresa^2 - b^2 = (a - b)(a + b)9x^2 - 25 = (3x - 5)(3x + 5)
Perfect squarea^2 + 2ab + b^2 = (a + b)^2x^2 + 10x + 25 = (x + 5)^2
Sum of cubesa^3 + b^3 = (a + b)(a^2 - ab + b^2)8x^3 + 27 = (2x + 3)(4x^2 - 6x + 9)
Difference of cubesa^3 - b^3 = (a - b)(a^2 + ab + b^2)x^3 - 8 = (x - 2)(x^2 + 2x + 4)

The AC method

Factor 6x^2 + x - 2: ac = -12; the numbers −3 and 4 multiply to −12 and add to 1.

\begin{aligned} 6x^2 + x - 2 &= 6x^2 - 3x + 4x - 2 \\ &= 3x(2x - 1) + 2(2x - 1) \\ &= (3x + 2)(2x - 1) \end{aligned}

Factoring by grouping

Four terms often pair up. In x^3 + 3x^2 - 4x - 12 the first two share x^2 and the last two share −4:

x^2(x + 3) - 4(x + 3) = (x + 3)(x^2 - 4) = (x + 3)(x - 2)(x + 2)

The bracket left over, x^2 - 4, is a difference of squares – factoring isn’t finished until no factor splits further. See the grouping in the calculator.

Why factor? Solving equations

A product is zero only when one of its factors is zero. Once x^3 + 3x^2 - 4x - 12 = 0 is written as (x + 3)(x - 2)(x + 2) = 0, the solutions can be read off: x = −3, 2 or −2. The same factors show where a graph crosses the x-axis and simplify algebraic fractions. Check any factoring by multiplying the brackets out again – you must get the original polynomial back.

When a polynomial cannot be factored with whole numbers it is called prime (irreducible over the rationals) – for example x^2 + 1 or x^2 - 2. To find the roots of such a quadratic, use the quadratic equation solver.

Frequently asked questions

What does it mean if a polynomial is prime?

It has no factors other than 1 and itself using whole-number coefficients, like x² + x + 1.