ganak.appFactoring Calculator
Free factoring calculator with step-by-step solutions. Factor quadratic, cubic, and quartic polynomials using the AC method, difference of squares, sum/difference of cubes, and rational root theorem. Also find all factors, prime factorization, and factor pairs of any number. No sign-up required.
x² + 5x + 6 =
(x + 3)(x + 2)
Factoring Details
Original, factored form & method breakdown
Step-by-Step Solution
How the expression was factored
Original expression
x² + 5x + 6
Identify the quadratic expression to factor.
AC Method
a × c = 1 × 6 = 6
Find two integers m, n where m × n = 6 and m + n = 5.
Found integers
m = 3, n = 2
3 × 2 = 6 ✔, 3 + 2 = 5 ✔
Write in factored form
(x + 3)(x + 2)
(x + 3)(x + 2) = x² + 5x + 6 ✔
What Is Factoring?
Breaking expressions into simpler multiplied parts
Factoring is the process of rewriting a mathematical expression as a product of simpler expressions. For example, the quadratic x² + 5x + 6 can be written as (x + 2)(x + 3). This is the reverse of expanding or multiplying out brackets.
For numbers, factoring means finding all integers that divide evenly into a given number. For instance, the factors of 48 are 1, 2, 3, 4, 6, 8, 12, 16, 24, and 48.
Key Idea
If A = B × C, then B and C are factors of A.
Common Factoring Methods
Six essential techniques for factoring polynomials
1. Greatest Common Factor (GCF)
Always start by factoring out the largest number and variable power common to all terms.
6x³ + 12x² + 18x = 6x(x² + 2x + 3)
2. Difference of Squares
When you have a² − b², it factors as (a + b)(a − b). Both terms must be perfect squares.
x² − 9 = (x + 3)(x − 3)
3. Perfect Square Trinomial
a² + 2ab + b² = (a + b)² and a² − 2ab + b² = (a − b)². Check if the middle term equals 2 × √(first) × √(last).
x² + 6x + 9 = (x + 3)²
4. Trinomial Factoring (AC Method)
For ax² + bx + c, find two numbers m and n where m × n = ac and m + n = b. This works for both a = 1 and a ≠ 1.
2x² + 7x + 3 = (2x + 1)(x + 3)
5. Sum & Difference of Cubes
a³ + b³ = (a + b)(a² − ab + b²) and a³ − b³ = (a − b)(a² + ab + b²). Remember the SOAP mnemonic: Same, Opposite, Always Positive.
x³ − 8 = (x − 2)(x² + 2x + 4)
6. Rational Root Theorem
For higher-degree polynomials, test possible rational roots ±(factors of constant) ÷ (factors of leading coefficient). When a root is found, use synthetic division to reduce the degree.
x³ − 6x² + 11x − 6 = (x − 1)(x − 2)(x − 3)
Worked Examples
Practice recognizing factoring patterns
| Expression | Factored Form | Method |
|---|---|---|
| x² + 7x + 12 | (x + 3)(x + 4) | Trinomial |
| x² − 16 | (x + 4)(x − 4) | Diff. of Squares |
| 4x² + 12x + 9 | (2x + 3)² | Perfect Square |
| 3x² + 10x − 8 | (3x − 2)(x + 4) | AC Method |
| x³ + 27 | (x + 3)(x² − 3x + 9) | Sum of Cubes |
| x³ − 4x² − 7x + 10 | (x − 1)(x + 2)(x − 5) | Rational Root |
Common Mistakes to Avoid
Pitfalls students frequently encounter
Forgetting to extract GCF first
Always check for a common factor before trying other methods. 6x² + 12x + 6 has a GCF of 6.
Sum of squares is NOT factorable
x² + 9 ≠ (x + 3)(x − 3). Only the DIFFERENCE of squares factors. x² + 9 is prime over the reals.
Sign errors in AC method
When ac is negative, one of m, n is positive and the other is negative. Double-check signs.
Not factoring completely
After one factoring step, check if any factor can be factored further. x⁴ − 16 = (x² + 4)(x² − 4) = (x² + 4)(x + 2)(x − 2).
Frequently Asked Questions
Common questions about factoring expressions and numbers