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A factoring calculator rewrites a polynomial as a product of simpler polynomials. For example, \(x^2 – 5x + 6 = (x – 2)(x – 3)\). This tool shows each step: the greatest common factor, special patterns, the rational root test, and a final check by multiplying back out.
How to Use the Factoring Calculator
- Type a polynomial, such as
x^3 - 6x^2 + 11x - 6. You can write3xinstead of3*x. - Press Solve.
- Read the steps to see which method was used. The Graph tab shows where the polynomial crosses the x-axis. Those points match the factors.
What Does It Mean to Factor?
Factoring is the reverse of multiplying. When you multiply \((x – 2)(x – 3)\), you get \(x^2 – 5x + 6\). Factoring starts with \(x^2 – 5x + 6\) and finds the pieces that were multiplied.
Factoring helps you solve equations, simplify fractions, and find limits. In calculus it is the first step of partial fraction decomposition and of “factor and cancel” in the limit calculator.
Factoring Methods, in the Order to Try Them
| Method | Use it when | Example |
|---|---|---|
| Greatest common factor (GCF) | Every term shares a factor | \(6x^3 + 9x^2 = 3x^2(2x + 3)\) |
| Difference of squares | Two perfect squares are subtracted | \(x^2 – 16 = (x – 4)(x + 4)\) |
| Perfect square trinomial | The first and last terms are squares | \(4x^2 – 12x + 9 = (2x – 3)^2\) |
| Trinomial (ac method) | \(ax^2 + bx + c\) | \(2x^2 + 7x + 3 = (2x + 1)(x + 3)\) |
| Sum or difference of cubes | Two perfect cubes | \(x^3 – 8 = (x – 2)(x^2 + 2x + 4)\) |
| Rational root theorem | Degree 3 or higher | \(x^3 – 6x^2 + 11x – 6\) |
Worked Examples
Example 1: \(6x^3 + 9x^2\)
Both terms are divisible by 3 and by \(x^2\). Pull out \(3x^2\): \(6x^3 + 9x^2 = 3x^2(2x + 3)\).
Example 2: \(2x^2 + 7x + 3\)
Multiply \(a \cdot c = 2 \cdot 3 = 6\). Find two numbers that multiply to 6 and add to 7: they are 6 and 1. Split the middle term: \(2x^2 + 6x + x + 3 = 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3)\).
Example 3: \(x^3 – 6x^2 + 11x – 6\)
By the rational root theorem, any rational root divides the constant 6, so try \(\pm1, \pm2, \pm3, \pm6\). Plug in \(x = 1\): \(1 – 6 + 11 – 6 = 0\). So \((x – 1)\) is a factor. Divide to get \(x^2 – 5x + 6\), which factors as \((x – 2)(x – 3)\).
Answer: \((x – 1)(x – 2)(x – 3)\).
Example 4: \(x^4 – 16\)
This is a difference of squares twice: \(x^4 – 16 = (x^2 – 4)(x^2 + 4) = (x – 2)(x + 2)(x^2 + 4)\). The last factor, \(x^2 + 4\), cannot be factored with real numbers.
Common Mistakes
- Skipping the GCF. Always look for a common factor first. It makes every later step easier.
- Factoring a sum of squares. \(x^2 + 9\) does not factor over the real numbers.
- Sign errors with cubes. \(a^3 – b^3 = (a – b)(a^2 + ab + b^2)\). The middle sign in the second factor is plus.
- Not checking. Multiply your factors back together. You should get the original polynomial.
Learn the Math Behind the Factoring Calculator
Factoring is the algebra skill calculus uses most: to cancel limits, find zeros and set up partial fractions.
Frequently Asked Questions
Can every polynomial be factored?
Every polynomial can be factored into linear and quadratic pieces with real numbers, but not always with whole numbers or fractions. When a piece has no rational roots, this calculator tells you it cannot be factored further over the rationals.
Can it factor expressions with two variables?
Yes. Type something like x^2 y - y. The calculator pulls out common factors and patterns in every variable.
How is factoring used in calculus?
You factor to cancel terms in limits, to split fractions with partial fractions before integrating, and to find where a derivative equals zero.
What is the rational root theorem?
If a polynomial with whole-number coefficients has a rational root \(\frac{p}{q}\), then \(p\) divides the constant term and \(q\) divides the leading coefficient. It gives a short list of numbers to test.
Related Guides
- Quadratic formula calculator for quadratics that do not factor nicely
- Simplify calculator for cancelling and combining terms
- Antiderivative practice problems