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Calculator Answers

Factoring Calculator

Blake Boege
Written by Blake Boege · Founder, Calculator Answers

A factoring calculator is an algebraic utility designed to break down polynomial expressions into their constituent prime factors. It evaluates quadratic equations, binomials, and trinomials, applying methods like finding the greatest common factor, factoring by grouping, and identifying special algebraic patterns like the difference of squares or perfect square trinomials. The calculator provides the simplified factored form along with the step-by-step mathematical reasoning. Students and algebra instructors use this tool to solve equations, simplify algebraic fractions, and verify homework assignments.

Paste a quadratic or linear expression in one variable. The calculator pulls out the greatest common factor, recognizes the difference of two squares, and finds the integer-root factoring of ax² + bx + c when it exists.

See also: Polynomial Calculator, Quadratic Formula Calculator, and Limit Calculator.

Quick Answer

Factor algebraic expressions and trinomials. Enter your polynomial function to see the factored form and step-by-step simplification steps.

Expression

Enter a quadratic or linear polynomial in one variable. Supported shapes: ax² + bx + c, ax² − b², and ax + b.

Use ^ for powers, * for explicit multiplication, and - for minus. · e.g. x^2 + 5x + 6

Single letter. · e.g. x

Supported methods

  • Pull out the greatest common factor from each term.
  • Recognize the difference of two squares: a²x² − b² = (ax − b)(ax + b).
  • Factor a quadratic with integer roots using the quadratic formula and reverse Vieta.

Irrational or complex roots are reported instead of being forced into a fake factoring.

Factored form

Factored form

(x + 2)(x + 3)

StepRoots from the quadratic formula are x = -2 and x = -3, so 1x² + 5x + 6 = (x − -2)(x − -3) after factoring out the leading coefficient.

Trying to factor a quadratic with irrational roots? Use the quadratic formula calculator for the exact roots, or the scientific calculator for a numeric approximation.

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Examples

x² + 5x + 6

(x − -2)(x − -3) = (x + 2)(x + 3)

2x² + 7x + 3

(2x − -1)(2x − -6) reduced to (2x + 1)(x + 3)

9x² − 16

(3x − 4)(3x + 4)

6x + 9

3(2x + 3)

How it works

The calculator parses the expression term by term and collects the coefficients of x², x, and the constant. From there it picks the right strategy.

GCF · ax + b = g · (a/g · x + b/g) where g = gcd(a, b)

Difference of squares · a²x² − b² = (ax − b)(ax + b)

Quadratic with integer roots · ax² + bx + c = a(x − r₁)(x − r₂)

Learn the concept

These guides explain the idea this tool computes, with worked examples and the distinctions readers most often trip over.

What Is a Coefficient in Math? Leading coefficients and why the sign belongs to the number.

What is factoring in algebra?

Factoring is the process of breaking down an algebraic expression into a product of simpler terms, or "factors," which when multiplied back together yield the original expression. It acts as the mathematical reverse of expanding expressions (such as using the FOIL method).

For example, if you expand the product (x + 2)(x + 3), you get x² + 5x + 6. Going in the opposite direction—starting with the trinomial and finding the binomial product—is factoring.

How to factor polynomials step-by-step

To successfully factor any algebraic expression, it is best to follow a structured hierarchy of strategies:

  1. Find the Greatest Common Factor (GCF): Examine all terms to see if they share a common numerical divisor or variable power. Pull it out first. For example, 4x² + 8x becomes 4x(x + 2).
  2. Count the Terms:
    • Two terms: Check if it fits a special pattern, like the Difference of Two Squares (a² − b²).
    • Three terms (Trinomial): If the expression is in the form ax² + bx + c, use the sum-product method (if a = 1) or the AC method (if a ≠ 1).
    • Four terms: Try factoring by grouping, splitting the polynomial into two pairs.
  3. Check if factors can be factored further: Always look at your binomial or trinomial results to ensure no further factoring (like another difference of squares) is possible.

Special factoring formulas to remember

Several algebraic identities appear frequently and can be factored instantly using standard templates:

  • Difference of Two Squares: a² − b² = (a − b)(a + b). Note that a Sum of Squares (a² + b²) cannot be factored using real numbers.
  • Perfect Square Trinomials: a² + 2ab + b² = (a + b)² and a² − 2ab + b² = (a − b)².
  • Sum of Two Cubes: a³ + b³ = (a + b)(a² − ab + b²).
  • Difference of Two Cubes: a³ − b³ = (a − b)(a² + ab + b²).

Worked example: Factoring a trinomial using the AC method

Let's factor the quadratic trinomial 3x² + 10x + 8 step-by-step.

Step 1: Identify coefficients

In the expression 3x² + 10x + 8, the coefficients are:
a = 3, b = 10, c = 8.

Step 2: Multiply 'a' and 'c' (Find AC)

AC = 3 × 8 = 24.

Step 3: Find two factors of AC that add to 'b'

We need two numbers that multiply to 24 and add to 10. Let's look at factor pairs of 24:
* 1 and 24 (adds to 25)
* 2 and 12 (adds to 14)
* 3 and 8 (adds to 11)
* 4 and 6 (adds to 10)
Our numbers are 4 and 6.

Step 4: Rewrite the middle term

Split the middle term 10x into 4x + 6x:
3x² + 6x + 4x + 8

Step 5: Factor by grouping

Group the first two terms and the last two terms:
* First group: (3x² + 6x) = 3x(x + 2)
* Second group: (4x + 8) = 4(x + 2)
Notice they both share a common binomial factor of (x + 2).

Step 6: Pull out the common binomial factor

Combine them to get the final factored form:
(3x + 4)(x + 2)

Common mistakes when factoring expressions

  • Forgetting the Greatest Common Factor: Jumping straight to factoring trinomials without checking if a GCF can be pulled out first. For example, factoring 2x² + 10x + 12 as (2x + 4)(x + 3) is less clean than pulling out 2 first to get 2(x² + 5x + 6) = 2(x + 2)(x + 3).
  • Sign errors with subtraction/negatives: When factoring by grouping, be extremely careful with signs. For example, in x² − 3x − 2x + 6, grouping the second half as −2(x − 3) is correct, whereas writing −2(x + 3) is a common sign error.
  • Trying to factor the sum of squares: Assuming x² + 9 can factor to (x + 3)(x + 3) or (x - 3)(x + 3). Remember that the sum of squares has no real factors.
  • Stopping too early: Forgetting to check if the factors themselves can be simplified further. E.g., factoring x⁴ − 16 as (x² − 4)(x² + 4) and failing to notice that x² − 4 factors further to (x − 2)(x + 2).

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Frequently asked questions

Factoring means breaking down an expression into a product of simpler terms or expressions (its factors). For example, factoring x² + 5x + 6 yields (x + 2)(x + 3). It is the reverse process of expanding or multiplying expressions.

It factors linear expressions (ax + b) by pulling out the greatest common factor (GCF), recognizes the difference of two squares (a²x² − b²), and finds binomial factors for quadratic trinomials (ax² + bx + c) with rational roots.

The GCF method involves identifying the largest factor that divides all terms in an expression. For example, in 6x² + 9x, the GCF of 6 and 9 is 3, and the GCF of x² and x is x. Factoring out 3x yields 3x(2x + 3). Check for a GCF first in every factoring problem.

For x² + bx + c, find two numbers that multiply to the constant 'c' and add to the coefficient 'b'. If you find two numbers, p and q, the factored form is (x + p)(x + q). For instance, x² + 7x + 12 factors to (x + 3)(x + 4) because 3 × 4 = 12 and 3 + 4 = 7.

For ax² + bx + c, multiply 'a' and 'c' to get 'ac'. Find two numbers that multiply to 'ac' and add to 'b'. Split the middle term 'bx' into two parts using these numbers, then factor by grouping. For example, to factor 2x² + 7x + 3, find two numbers multiplying to 6 (2×3) and adding to 7 (6 and 1). Rewrite as 2x² + 6x + x + 3 and group terms to get (2x + 1)(x + 3).

A quadratic ax² + bx + c won't factor over integers if its discriminant (b² − 4ac) is not a perfect square. If the discriminant is negative, it has complex roots. If it is positive but not a perfect square, it has irrational roots. This calculator only returns factored forms with rational coefficients.

The Difference of Two Squares identity is a² − b² = (a − b)(a + b). In algebra, an expression like 4x² − 9 can be rewritten as (2x)² − 3², which factors directly to (2x − 3)(2x + 3). There is no real-number factoring formula for the sum of two squares (a² + b²).

Factoring allows you to use the Zero Product Property to solve equations. If an expression is set to zero, e.g., (x − 2)(x + 3) = 0, then either x − 2 = 0 or x + 3 = 0, giving the roots x = 2 and x = −3. Factoring is often the fastest way to solve polynomial equations.