Polynomial Solver (Degree ≤ 5)
Solve Polynomial Equations Up to Degree 5
Polynomial equations are foundational in algebra and calculus. From designing roller coasters to modeling population growth, polynomials help us represent real-world relationships using algebraic expressions. But solving higher-degree polynomial equations—like cubic, quartic, or quintic—can be tricky without the right tools.
That’s where our Polynomial Solver comes in.
Whether you’re a student, teacher, engineer, or someone brushing up on algebra, our free online Polynomial Equation Solver helps you solve polynomial expressions up to degree 5 in a clear, structured, and easy-to-understand way—with detailed, step-by-step solutions.
What is a Polynomial Equation?
A polynomial is an expression consisting of variables, coefficients, and exponents that are whole numbers. A polynomial equation sets such an expression equal to zero. Here’s a general form: anxn+an−1xn−1+⋯+a2x2+a1x+a0=0a_nx^n + a_{n-1}x^{n-1} + \dots + a_2x^2 + a_1x + a_0 = 0an
- The degree of the polynomial is the highest power of xxx with a non-zero coefficient.
- The roots or solutions are the values of xxx that make the equation true.
Our solver supports equations from linear (degree 1) all the way up to quintic (degree 5).
Why Solving Polynomials Matters
Polynomials aren’t just abstract math—they model countless practical situations, such as:
- Physics: Motion, wave behavior, and trajectory paths
- Economics: Supply/demand curves, profit maximization
- Biology: Modeling population growth or decay
- Engineering: Calculating loads, forces, and stability
- Computer Graphics: Rendering curves and animations
Understanding how to solve them accurately is key to solving problems in school and in real life.
Common Types of Polynomial Equations You Can Solve
Our calculator walks you through all these, with explanations along the way.
Key Features of the Polynomial Solver
- Supports Up to Degree 5: Handle anything from simple linear to complex quintic equations.
- Step-by-Step Solutions: Learn how to solve, not just the answer.
- Method Selection: Choose your solving method—factoring, Rational Root Theorem, synthetic division, or quadratic formula.
- Real-Time Feedback: Get instant results and validation for your expressions.
- User-Friendly Interface: No clutter—just the input, your selected method, and clear results..
How to Use the Solver
Enter your polynomial in the input box.
Example: x^4 - 3x^3 + 2x - 6
Choose a solving method, such as:
- Rational Root Theorem
- Synthetic Division
- Factorization
- Quadratic Formula (auto-selected for degree 2)
- Click “Solve” and watch the solution unfold with detailed explanation steps.
- Review the result in the output box, which includes:
- Detected degree
- Roots or factored form
- Every intermediate step
It’s that easy!
Methods definition
🟡 Rational Root Theorem
This method lists all possible rational roots based on the constant term and leading coefficient. It’s especially useful for cubic, quartic, and quintic polynomials.
🟢 Synthetic Division
Once a root is guessed or detected, synthetic division simplifies the polynomial by dividing it, reducing the degree for further solving.
🔵 Factoring
Factoring breaks the polynomial into simpler binomials or quadratics. Works great when roots are integers or recognizable patterns exist.
🟣 Quadratic Formula
Used for quadratic (degree 2) equations to find real or complex roots. Also used in the final step of solving reduced cubic or quartic equations.