How the polynomial is evaluated
You enter the coefficients from the highest power down to the constant term, so the list 1, -3, 2 represents x² − 3x + 2. The tool reads them in that order and plugs in your chosen value of x.
Rather than computing each power separately, it uses Horner's method, which rewrites the polynomial as a nested set of multiply-then-add steps. This is faster and far more numerically stable, especially for high degrees or large values of x.
Reading the result
The value at x is the result of substituting your number into the polynomial. The degree is simply the highest power present, which equals the number of coefficients minus one.
Roots — the x-values where the polynomial equals zero — are reported for linear and quadratic polynomials. For a quadratic, the discriminant b² − 4ac decides the picture: positive gives two real roots, zero gives one repeated root, and negative gives a conjugate pair of complex roots.
- Value at x: the polynomial evaluated at your input.
- Degree: the highest power, one less than the coefficient count.
- Roots: shown for degree 1 and degree 2 only.
Practical tips
Order and completeness of coefficients matter. Every power between the highest and the constant must be represented, even when its coefficient is zero.
- Include a 0 for any missing term, e.g. x³ + 1 is 1, 0, 0, 1.
- A value of zero at x means that x is a root of the polynomial.
- Coefficients can be separated by commas or spaces.
Common mistakes
The two recurring errors are entering the coefficients in the wrong direction and dropping zero terms. Both quietly change which polynomial you are actually evaluating.
- List coefficients highest power first, not lowest.
- Do not omit zero coefficients for missing powers.
- A leading coefficient of zero is not allowed when solving for roots.
Formula
p(x) = cₙxⁿ + … + c₁x + c₀; quadratic roots = (−b ± √(b²−4ac)) / 2aFrequently asked questions
- Why are roots not shown for cubics?
- Closed-form root finding is only included for linear and quadratic polynomials. Higher degrees report the value at x and the degree only.