Factoring Calculator: Polynomials, Steps
This factoring calculator factors polynomials in one variable completely over the integers and shows the steps. Type an expression such as x^2 + 5x + 6, 2x^2 + 7x + 3, x^4 - 16, 6x^3 - 9x^2 or x^3 + 8, and it pulls out the greatest common factor, finds every rational root with the rational root theorem, splits off the matching linear factors, and checks the remaining part for quadratic factors. It handles trinomials with a leading coefficient, difference of squares, sum and difference of cubes and higher powers up to degree 8. Every answer is multiplied back out to confirm it matches your polynomial, and the real roots are listed too, which makes it handy for algebra homework and test review.
Factoring Calculator
Factor polynomials completely over the integers: GCF, trinomials, difference of squares, cubes and more.
How to use the Factoring Calculator
- Type a polynomial, like 2x^2 + 7x + 3. Use ^ for powers.
- Read the fully factored form at the top.
- Check the steps: GCF, rational roots and any quadratic factors.
- Use the roots and the expanded check to verify your work.
Factoring patterns
GCF first: 6x^3 - 9x^2 = 3x^2(2x - 3). Trinomial: x^2 + (a+b)x + ab = (x + a)(x + b). Difference of squares: a^2 - b^2 = (a - b)(a + b). Cubes: a^3 ± b^3 = (a ± b)(a^2 ∓ ab + b^2).
Rational root theorem: any rational root p/q has p dividing the constant term and q dividing the leading coefficient. Each root r = p/q gives a factor (qx - p).
Features
- Factors completely over the integers
- Greatest common factor first
- Trinomials, including a leading coefficient
- Difference of squares and sum or difference of cubes
- Quartics that split into two quadratics
- Answer checked by expanding it back
Frequently asked questions
How do you factor a trinomial like x^2 + 5x + 6?
Find two numbers that multiply to 6 and add to 5. They are 2 and 3, so x^2 + 5x + 6 = (x + 2)(x + 3).
How do you factor a difference of squares?
Use a^2 - b^2 = (a - b)(a + b). For example x^2 - 49 = (x - 7)(x + 7), and x^4 - 16 = (x - 2)(x + 2)(x^2 + 4).
What does "cannot be factored over the integers" mean?
The polynomial has no factors with whole-number coefficients, such as x^2 + 1 or x^2 - 2. It is called prime or irreducible. It may still have real or complex roots.
How do I factor the sum of cubes?
Use a^3 + b^3 = (a + b)(a^2 - ab + b^2). For example x^3 + 8 = (x + 2)(x^2 - 2x + 4).
Can it factor with decimals or two variables?
It factors one-variable polynomials with whole-number coefficients. Multiply through to clear decimals or fractions first, for example 0.5x^2 + x becomes x^2 + 2x.