purpose
Polynomial Root Finder
Find all roots of any polynomial up to degree 10. Enter the expression and see real and complex roots with interactive graph.
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Try an Example
Pick a scenario to see how the calculator works, then adjust the values
Quadratic Equation
Solve a classic quadratic with two real roots.
Key values: x^2 - 5x + 6 = 0 · Two real roots
Linear Equation
Solve a simple linear equation for x.
Key values: 3x + 7 = 22 · Single root
Transcendental Equation
Find roots of sin(x) = x/3 using numerical methods.
Key values: sin(x) = x/3 · Numerical solution
Cubic with Complex Roots
Solve a cubic equation and show complex roots.
Key values: x^3 + x + 1 = 0 · Complex roots enabled
Polynomial Classification
A polynomial of degree has the form:
| Degree | Name | Max real roots | Solution method |
|---|---|---|---|
| 1 | Linear | 1 | Direct algebra |
| 2 | Quadratic | 2 | Quadratic formula |
| 3 | Cubic | 3 | Cardano's formula |
| 4 | Quartic | 4 | Ferrari's method |
| ≥ 5 | Quintic+ | n | Numerical methods only |
Fundamental Theorem of Algebra
Every polynomial of degree with complex coefficients has exactly roots in (counting multiplicity). This means:
Real polynomial corollary: Complex roots of real polynomials always come in conjugate pairs. So a degree-3 real polynomial always has at least one real root.
Rational Root Theorem
If a polynomial with integer coefficients has a rational root (in lowest terms), then:
- divides the constant term
- divides the leading coefficient
This gives a finite list of candidates to test. For example, for :
Synthetic Division
Once you find one root , use synthetic division to factor out and reduce the degree by 1. This is faster than long division for linear divisors.
Strategy for higher-degree polynomials: Use the rational root theorem to find one root → synthetic division to reduce the degree → repeat until you reach a quadratic, which you solve with the quadratic formula.
Frequently Asked Questions
What is the Fundamental Theorem of Algebra?
The Fundamental Theorem of Algebra states that every polynomial of degree has exactly roots in the complex numbers, counting multiplicity. This means a degree-4 polynomial always has four roots, though some may be complex.
Can all polynomials be solved with a formula?
No. The Abel-Ruffini theorem proves that no general algebraic formula exists for polynomials of degree 5 or higher. Polynomials up to degree 4 have exact formulas (linear, quadratic, Cardano, Ferrari). Degree 5 and above require numerical methods.
What is the Rational Root Theorem?
The Rational Root Theorem states that if a polynomial with integer coefficients has a rational root (in lowest terms), then divides the constant term and divides the leading coefficient. This gives a finite list of candidates to test.
What is synthetic division?
Synthetic division is a shorthand method for dividing a polynomial by a linear factor . Once you find one root , synthetic division factors out and reduces the polynomial degree by one, making it easier to find remaining roots.
Why do complex roots come in conjugate pairs?
For polynomials with real coefficients, complex roots always appear in conjugate pairs ( and ). This is because the complex conjugate of a real polynomial evaluated at equals the polynomial evaluated at the conjugate of .
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