Factoring Calculator
Use this factoring calculator to factor a whole number into its prime factors and factor list, or factor a quadratic ax² + bx + c into binomials with its roots.
Factoring Calculator
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About This Factoring Calculator
This factoring calculator works two ways. In whole-number mode it lists every factor of an integer and its prime factorization. In quadratic mode it factors an expression of the form ax² + bx + c into binomials and shows the roots and discriminant.
It is useful for number theory practice, simplifying fractions, and algebra homework where a quadratic needs to be factored before it can be solved.
Factoring Whole Numbers
Factoring a whole number means finding the numbers that divide it exactly. The prime factorization goes further, writing the number as a product of prime numbers only.
For example, the factors of 60 are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60. Its prime factorization is 2² × 3 × 5.
Factor Pairs
Factors come in pairs that multiply to the original number. For 60 the factor pairs are 1×60, 2×30, 3×20, 4×15, 5×12, and 6×10.
Prime and Composite Numbers
A prime number has exactly two factors: 1 and itself. A composite number has more. For example, 97 is prime, while 60 is composite. The number 1 is neither prime nor composite.
Factoring Quadratics
A quadratic has the form ax² + bx + c. Factoring rewrites it as a product of binomials. The calculator finds the roots with the quadratic formula and turns each root into a factor.
The formula is:
ax² + bx + c = a(x − r₁)(x − r₂), where r = (−b ± √(b² − 4ac)) ÷ 2a
Example
Take x² + 5x + 6. The discriminant is 5² − 4·1·6 = 1, a perfect square, so the roots are rational: x = -2 and x = -3. Each root r gives a factor (x − r), so:
x² + 5x + 6 = (x + 2)(x + 3)
The Discriminant
The discriminant is b² − 4ac, and it tells you what kind of factors to expect:
| Discriminant | What It Means |
|---|---|
| Perfect square | Rational roots — factors with integer coefficients |
| Positive, not a perfect square | Irrational roots — factors use decimals |
| Zero | One repeated root |
| Negative | No real factors — the roots are complex |
How to Use the Calculator
- Choose whole-number or quadratic mode.
- For a whole number, enter the integer.
- For a quadratic, enter the coefficients a, b, and c.
- Click calculate.
- Review the factors, roots, and steps.
When Should You Use This Tool?
- Finding the prime factorization of a number
- Listing all factors or factor pairs
- Checking whether a number is prime
- Factoring a quadratic before solving it
- Checking algebra homework
Related Calculators
You may also find these tools useful:
- Square Root Calculator
- Simplify Calculator
- Prime Factorization Calculatorcoming soon
- GCF Calculatorcoming soon
- LCM Calculatorcoming soon
- Quadratic Formula Calculatorcoming soon
Start Calculating
Enter a whole number or a quadratic above and use the factoring calculator to find the factors, prime factorization, or binomial form with steps.
Frequently Asked Questions
What does this factoring calculator do?
It factors whole numbers and quadratic expressions. For a whole number it lists every factor and the prime factorization. For a quadratic it finds the binomial factors, the roots, and the discriminant.
What is prime factorization?
Prime factorization writes a number as a product of prime numbers. For example, 60 = 2² × 3 × 5. Every whole number greater than 1 has a unique prime factorization.
How do you factor a quadratic?
Find the roots of ax² + bx + c using the quadratic formula. Each root r gives a factor (x − r). For example, x² + 5x + 6 has roots -2 and -3, so it factors as (x + 2)(x + 3).
What is the discriminant?
The discriminant is b² − 4ac. When it is a perfect square the quadratic factors with integer coefficients, when it is positive but not a perfect square the roots are irrational, and when it is negative there are no real factors.
Can every quadratic be factored?
Every quadratic can be written in factored form using its roots, but it only factors neatly into integer binomials when the discriminant is a perfect square. A negative discriminant means it does not factor over the real numbers.
Is 1 a prime number?
No. 1 is neither prime nor composite. It has a single factor — itself — and no prime factorization.
What is a factor pair?
A factor pair is two numbers that multiply to give the original number. For example, 60 has the factor pairs 1×60, 2×30, 3×20, 4×15, 5×12, and 6×10.