# What does factoring a polynomial give?

Factors a polynomial over the rational numbers, checking the product of the factors.

- Page: https://www.acalculator.org/math/factoring-calculator
- JSON spec: https://www.acalculator.org/math/factoring-calculator.json
- Version: 191af2d945e1

## Default answer

Example with the default inputs (Polynomial x^2 + 2x - 15): Factoring x^2 + 2x - 15 gives (x - 3) (x + 5).

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| p | Polynomial | The polynomial, in one letter. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| factored | Factored form | The polynomial as a product of factors. |

## Method

A computer algebra system factors over the rationals; the factors show only when their product equals the polynomial at 20 points.

## Assumptions

- Factors have rational coefficients.
- An answer that fails its check is not shown.

## Worked examples

1. p = x^2 + 2x - 15 gives factored = (x - 3) (x + 5). Source: OpenStax College Algebra 2e, 1.5, Ex. 2. https://openstax.org/books/college-algebra-2e/pages/1-5-factoring-polynomials.
2. p = 5x^2 + 7x - 6 gives factored = (5x - 3) (x + 2).
3. p = 8x^3 - 125 gives factored = (2x - 5) (4x^2 + 10x + 25).

## FAQ

### What does factoring a polynomial mean?

Writing it as a product of polynomials of lower degree. x² + 2x − 15 = (x − 3)(x + 5), because (x − 3)(x + 5) = x² + 5x − 3x − 15. Factored form shows the roots at once: the product is 0 exactly when x = 3 or x = −5.

### How do you factor a trinomial by hand?

For x² + bx + c, find two numbers that multiply to c and add to b: for x² + 2x − 15 they are −3 and 5. For ax² + bx + c with a ≠ 1, find two numbers that multiply to ac and add to b, split the middle term, and factor by grouping: 5x² + 7x − 6 = 5x² + 10x − 3x − 6 = 5x(x + 2) − 3(x + 2) = (5x − 3)(x + 2).

### Why is x² − 2 not factored?

The page factors over the rational numbers: every coefficient in a factor is a whole number or a fraction. x² − 2 = (x − √2)(x + √2) needs √2, which is irrational, so over the rationals x² − 2 is prime (irreducible) and is shown unchanged. The same goes for x² + 1, which has no real roots at all.

### Which special patterns does it use?

The standard ones and more: a difference of squares a² − b² = (a − b)(a + b), perfect squares a² ± 2ab + b² = (a ± b)², and sums and differences of cubes a³ ± b³ = (a ± b)(a² ∓ ab + b²). It also pulls out a common factor first: 2x³ + 4x² + 2x = 2x(x + 1)².

### Can I factor a number?

This page factors polynomials, so a plain number such as 12 gets no answer here. To break a whole number into primes, use the factor calculator.

### What does "No verified answer" mean?

The factors are multiplied back out, in effect: the page compares their product with your polynomial at 20 points. If they differ anywhere, or no factoring is found, or the work takes over 3 seconds, the page says "No verified answer" instead of showing factors it could not check.

## Sources

- OpenStax, College Algebra 2e, section 1.5 Factoring Polynomials: https://openstax.org/books/college-algebra-2e/pages/1-5-factoring-polynomials
- OpenStax, Intermediate Algebra 2e, section 6.4 General Strategy for Factoring Polynomials: https://openstax.org/books/intermediate-algebra-2e/pages/6-4-general-strategy-for-factoring-polynomials