ALGEBRA / POLYNOMIAL

Polynomial Equation Calculator for Real and Complex Roots up to Degree Six

Updated October 13, 2026By Unit Conversion Plus

Find approximate real and complex roots of polynomials up to degree six, with repeated roots, residual checks and a graph.

Set up your equation

aₙxⁿ + … + a₁x + a₀ = 0

How to use this solver

List coefficients from the highest power down to the constant, including zeros for every missing power. For x³ − 4x + 2 = 0, enter 1, 0, −4, 2. Repeated factors are separated using exact arithmetic before a numerical solver finds their roots. The real-valued graph shows x-axis crossings, while complex roots appear in the results.

Examples and checks for polynomial equations

Can I calculate a uniform scale factor from a volume change?

For geometrically similar objects, volume scales with the cube of the linear scale factor. If the new volume is 27 times the original, solve x³ − 27 = 0 by entering 1, 0, 0, −27. The positive real root is approximately 3, so every length is multiplied by 3. This relationship assumes all dimensions scale by the same factor.

Why do I get more roots than usable answers?

The solver considers complex numbers as well as real numbers, and reports roots without knowing your physical restrictions. Lengths may need to be positive, times may need to lie within a measured interval, and cut sizes must leave room for the remaining material. Substitute each candidate into the original problem and apply those restrictions before choosing an answer.

What does a repeated root mean in a real-world model?

A repeated root has multiplicity greater than one. At an even-multiplicity real root, the polynomial touches zero without changing sign nearby. In the 12 cm sheet example, the repeated root at x = 2 corresponds to the maximum box volume of 128 cm³. Repeated roots can be sensitive: a small change in the target volume can change the number of usable real solutions.

Can I paste measured data to create a polynomial model?

No. This calculator needs coefficients of an equation you already have. Use a regression or curve-fitting tool to estimate those coefficients from measurements, then solve for a target value here. A small root-check residual means the root fits the entered polynomial; it does not show that the polynomial accurately predicts real measurements.

How precise are the answers, and should I trust every displayed digit?

Roots are numerical approximations. The calculator checks convergence and a scaled residual, but these checks do not guarantee that every displayed digit is accurate, especially for closely spaced roots or poorly scaled coefficients. Keep enough precision when checking the equation, then report a final answer consistent with the accuracy of your original measurements. Avoid using a model beyond its intended range.

Interpret roots together with multiplicity and residuals

The equation x³ − 4x² + 5x − 2 = 0 factors as (x − 1)²(x − 2). It has three roots when multiplicity is counted, but only two distinct real values. Substitution error indicates how closely an approximate root satisfies the polynomial; a small residual alone does not guarantee many accurate digits for an ill-conditioned or repeated root.

Frequently asked questions

Does this show symbolic factoring steps?

It explains the numerical method and checks the roots, rather than claiming a symbolic factorisation. The output gives approximate roots and their multiplicities.

What are the limits?

Enter 2–7 coefficients for degree one through six. Very poorly scaled coefficients or closely spaced roots can be numerically difficult; the calculator reports a failure if its convergence or residual checks do not pass.

Can I type a whole equation?

Use the labelled coefficient fields or coefficient list. First collect like terms and expand brackets yourself. This solver does not parse unrestricted equation text.

How can I find the dimensions of a box from its volume?

Suppose a rectangular box has dimensions x, x + 1 and x + 2 centimetres, and its volume is 120 cm³. Expanding x(x + 1)(x + 2) = 120 gives x³ + 3x² + 2x − 120 = 0. Enter 1, 3, 2, −120. The real root is approximately 4, giving dimensions 4 cm, 5 cm and 6 cm. The other two roots are complex and do not represent physical lengths.

Can I calculate the corner cut size for an open-top box?

Suppose you cut squares of side x cm from each corner of a 12 cm by 12 cm sheet, then fold up the sides. The volume is x(12 − 2x)². For a target volume of 128 cm³, enter 4, −48, 144, −128. The roots are approximately 2, with multiplicity 2, and 8. Only x = 2 cm is valid because the cut size must be greater than 0 and less than 6 cm. The resulting box is 8 cm by 8 cm by 2 cm.

How do I find when a model reaches a target value?

Move the target value to the left before entering coefficients. Suppose a model is f(t) = t³ − 9t² + 23t + 5, and you want f(t) = 20. Enter 1, −9, 23, −15. The roots are approximately t = 1, 3 and 5. They are candidate times when the model reaches 20; retain only times within the interval where your model applies.