Polynomial Factoring Calculator

Polynomial Factoring:

\[ P(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_0 = a_n(x - r_1)(x - r_2)\cdots(x - r_n) \]

Factored Form:

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1. What is Polynomial Factoring?

Factoring polynomials means expressing them as a product of simpler polynomials. For example, x² - 5x + 6 factors into (x - 2)(x - 3). This reveals the roots of the polynomial (values of x that make the polynomial equal to zero).

2. How Does Factoring Work?

The fundamental theorem of algebra states that every non-zero single-variable polynomial of degree n has exactly n roots (including complex roots). The factored form is:

\[ P(x) = a_n(x - r_1)(x - r_2)\cdots(x - r_n) \]

Where:

\( a_n \) — Leading coefficient
\( r_1, r_2, \ldots, r_n \) — Roots of the polynomial

Explanation: For quadratic polynomials (degree 2), we can use the quadratic formula. Higher degree polynomials may require numerical methods.

3. Importance of Factoring

Details: Factoring is essential for solving polynomial equations, analyzing graphs of functions, simplifying rational expressions, and in many areas of mathematics and engineering.

4. Using the Calculator

Tips: Enter the polynomial coefficients from highest degree to lowest, separated by commas. For example, for x² - 5x + 6, enter "1,-5,6".

5. Frequently Asked Questions (FAQ)

Q1: What if my polynomial has complex roots?
A: The calculator will display complex roots in the form a ± bi when they occur.

Q2: Can this factor any degree polynomial?
A: Currently optimized for quadratics. Higher degrees may require specialized algorithms.

Q3: What about multiple roots?
A: Multiple roots appear as repeated factors, like (x - r)².

Q4: How precise are the roots?
A: Roots are displayed with 4 decimal places for clarity.

Q5: Can I factor polynomials with irrational coefficients?
A: Yes, but exact factoring may not always be possible numerically.