Polynomial Roots Calculator
Find the roots of polynomial equations quickly and accurately. Enter polynomial coefficients to solve linear, quadratic, cubic, or quartic equations.
Find Polynomial Roots
What Are Polynomial Roots?
A root of a polynomial is a value of x that makes the polynomial equal to zero. In other words, if f(x) is a polynomial, a root is a value that satisfies f(x) = 0.
For x² − 5x + 6 = 0, the roots are x = 2 and x = 3.
How to Use This Calculator
Enter the coefficients of your polynomial equation. Coefficients that are not needed can be left as zero.
- Enter the x⁴ coefficient.
- Enter the x³ coefficient.
- Enter the x² coefficient.
- Enter the x coefficient.
- Enter the constant.
- Click Calculate Polynomial Roots.
Polynomial Equation Form
A polynomial can contain terms with different powers of x. This calculator accepts equations up to degree four.
ax⁴ + bx³ + cx² + dx + e = 0
If the x⁴ coefficient is zero, the calculator automatically treats the equation as a cubic, quadratic, or linear equation depending on the highest non-zero coefficient.
Quadratic Polynomial Roots
A quadratic polynomial has the form ax² + bx + c = 0. Its roots can be real or complex, depending on the discriminant.
x² − 5x + 6 = 0
Roots: 2 and 3
Complex Polynomial Roots
Some polynomial equations do not have real roots. Their solutions can instead be complex numbers containing the imaginary unit i.
x² + 1 = 0
Roots: i and −i
Polynomial Degree and Number of Roots
The degree of a polynomial is determined by its highest power of x with a non-zero coefficient. A polynomial of degree n has exactly n roots in the complex numbers, counting repeated roots according to their multiplicity.
- Degree 1: up to 1 root
- Degree 2: up to 2 roots
- Degree 3: up to 3 roots
- Degree 4: up to 4 roots