Factoring Calculator
Factor integers, quadratic expressions, and polynomials. Prime factorization with all divisors, quadratic factoring using roots, binomial expansion, fraction simplification, and polynomial long division.
Quick Calculator Get a fast estimate
Prime Factorization
2^3 × 3^2 × 5
Number of divisors
24
All divisors (24)
1234568910121518202430364045607290120180360
How to Use the Factoring Calculator
For integer factoring, enter the number to get prime factorization and all divisors. For quadratic factoring, enter coefficients a, b, c to get the factored form a(x−r₁)(x−r₂).
The Extended Calculator adds binomial expansion and fraction simplification. The Professional Calculator handles general polynomial long division.
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Extended Calculator More options, charts, and scenario comparison
Factor: ax² + bx + c
Factored Form
(x − 4)(x − 3)
Root x₁
4
Root x₂
3
Vertex form
1(x−3.5)²+-0.25
Factoring Identities
Difference of squares: a²−b² = (a+b)(a−b)
Perfect square: a²+2ab+b² = (a+b)²
a²−2ab+b² = (a−b)²
Sum of cubes: a³+b³ = (a+b)(a²−ab+b²)
Difference of cubes: a³−b³ = (a−b)(a²+ab+b²)
Quadratic (via formula): ax²+bx+c = a(x−r₁)(x−r₂)
where r₁,r₂ = (−b ± √(b²−4ac)) / (2a)
Factoring Strategy
| Step | Check | Method |
|---|---|---|
| 1 | Common factor? | Factor out GCF |
| 2 | 2 terms? | Difference of squares, sum/diff of cubes |
| 3 | 3 terms (a=1)? | Find factors of c that sum to b |
| 4 | 3 terms (a≠1)? | AC method or quadratic formula |
| 5 | 4 terms? | Group factoring |
Need full precision?
Professional Calculator Complete parameters, sensitivity analysis, and detailed breakdown
Polynomial Long Division
Dividend: x^3-6x^2+11x-6
Divisor: x-2
Quotient
x^2-4x+3
Remainder
0
Exact division?
Yes
Result: (x^3-6x^2+11x-6) = (x-2) × (x^2-4x+3) + (0)
Step-by-step (3 steps)
- Quotient term: 1 | Subtract 1×(x-2)
- Quotient term: -4 | Subtract -4×(x-2)
- Quotient term: 3 | Subtract 3×(x-2)
Frequently Asked Questions
Find roots using the quadratic formula, then write in factored form. For x²−5x+6=0: roots are x=2 and x=3, so x²−5x+6=(x−2)(x−3). Verify by expanding: x²−3x−2x+6=x²−5x+6 ✓
a²−b²=(a+b)(a−b). Examples: x²−25=(x+5)(x−5), 4x²−9=(2x+3)(2x−3), 100−y²=(10+y)(10−y). Check: multiply the factors to verify you get back the original.
Divide by 2 until odd, then try 3, 5, 7... up to √n. Example: 360=2×180=4×90=8×45=8×9×5=2³×3²×5. The number of divisors = (3+1)(2+1)(1+1) = 24.
Divide the leading term of the dividend by the leading term of the divisor, multiply the divisor by this quotient term, subtract, then repeat with the remainder. Used to factor higher-degree polynomials.
(a+b)ⁿ = Σₖ C(n,k)aⁿ⁻ᵏbᵏ. The coefficients are from row n of Pascal's triangle. Example: (x+2)⁴ = x⁴+8x³+24x²+32x+16.