Polynomial Root Calculator – Real and Complex Roots
Find real and complex roots of polynomial equations from degree 1 through degree 6. Enter the coefficients and calculate numerical solutions online.
Use this polynomial root calculator to find the numerical values of x that make a polynomial equal to zero. It supports real roots, complex roots, and repeated roots for equations from degree 1 through degree 6.
How to Use the Polynomial Root Calculator
Select the degree of the polynomial and enter the coefficient of every term from the highest power down to the constant. Include zero coefficients for any powers that do not appear in the equation, then press Calculate Roots.
What Is a Polynomial Root?
A polynomial root, also called a zero, is a value of x for which the polynomial equals zero. For example, x = 2 is a root of x² - 4 because substituting 2 gives 2² - 4 = 0.
Why the Leading Coefficient Cannot Be Zero
The coefficient of the highest selected power must be nonzero. If it is zero, the equation has a lower degree than the one selected. For example, 0x³ + x² - 1 is quadratic, not cubic.
Enter Zero for Missing Polynomial Terms
Every power between the selected degree and the constant term needs a coefficient. If a term is absent, enter 0. For example, x³ + 2x - 5 has coefficients 1, 0, 2, and -5 because the x² term is missing.
Real and Complex Polynomial Roots
Some polynomial equations have real roots, while others include complex roots. Complex results are displayed with i as the imaginary unit, where i² = -1. Polynomials with real coefficients produce nonreal complex roots in conjugate pairs.
Repeated Roots and Multiplicity
A root may occur more than once. For example, x² - 2x + 1 equals (x - 1)², so x = 1 is a repeated root with multiplicity 2. Repeated roots may appear more than once in the result list.
Supported Polynomial Degrees
The calculator supports linear, quadratic, cubic, quartic, quintic, and sextic equations, covering polynomial degrees 1 through 6.
Numerical Root Approximations
Roots are calculated numerically and displayed with up to 10 decimal places. Irrational and complex solutions may therefore be rounded approximations rather than exact symbolic expressions.
Polynomial Root Calculator Example
To solve x³ - 6x² + 11x - 6 = 0, select Cubic — Degree 3 and enter 1, -6, 11, and -6. The calculator returns the three roots 1, 2, and 3.
Frequently Asked Questions
What does a polynomial root calculator do?
It finds the values of x that make a polynomial equation equal to zero. This calculator returns numerical real and complex roots for degrees 1 through 6.
What polynomial degrees are supported?
The calculator supports linear, quadratic, cubic, quartic, quintic, and sextic polynomials, covering degrees 1 through 6.
Can this calculator find complex roots?
Yes. Complex roots are displayed using i as the imaginary unit. For example, the roots of x² + 1 are i and -i.
Can it solve cubic and quartic equations?
Yes. It calculates numerical roots for cubic equations of degree 3 and quartic equations of degree 4.
Can it solve quintic and sextic equations?
Yes. Numerical roots are supported for quintic equations of degree 5 and sextic equations of degree 6.
What should I enter for a missing term?
Enter 0 as its coefficient. For example, x³ + 2x - 5 requires the coefficients 1, 0, 2, and -5.
Can the leading coefficient be zero?
No. The highest-degree coefficient must be nonzero. If it is zero, select the lower degree that matches the actual polynomial.
Can a polynomial have repeated roots?
Yes. A repeated root occurs more than once and has a multiplicity greater than 1. It may appear multiple times in the result list.
Are polynomial roots always real?
No. Polynomial roots may be real or complex. A degree-n polynomial has n complex roots when repeated roots are counted according to multiplicity.
Are the displayed roots exact?
Not always. The calculator uses numerical methods and displays up to 10 decimal places, so irrational and complex roots may be rounded.
Why could the calculator fail for some coefficients?
Numerical root finding can become unstable for extreme coefficient sizes or closely clustered roots. In that case, the calculator reports that stable roots could not be calculated instead of showing invalid values.
Is the polynomial root calculator free?
Yes. It is free to use and does not require an account.