Polynomial Roots Calculator
A polynomial root is any value of x that makes the polynomial equal to zero, and finding those roots is one of the most fundamental tasks in algebra. This calculator solves polynomials up to degree three: linear equations of the form cx plus d, quadratic equations of the form bx squared plus cx plus d, and cubic equations of the form ax cubed plus bx squared plus cx plus d. You enter four coefficients, a, b, c, and d, leaving any unused coefficient at zero, and the calculator returns every real root along with the root type and, for quadratics, the discriminant. For a linear equation the single root is just minus d divided by c. For a quadratic the calculator evaluates the discriminant b squared minus 4ac: a positive discriminant gives two distinct real roots found by the quadratic formula, zero gives one repeated real root, and a negative discriminant means no real roots exist. For a cubic the calculator applies Cardano's method, first reducing the equation to a depressed cubic by substitution, then classifying the roots using the cubic discriminant. Three distinct real roots are found using trigonometric identities; a repeated root is returned when the discriminant is zero; and one real root plus two complex conjugate roots is reported when the discriminant is negative. The calculator handles any combination of signed decimal coefficients and automatically detects the effective degree. It suits students working through algebra problems, engineers solving characteristic equations, and anyone verifying hand-calculated roots. Results are numerical values rounded to four decimal places.
Only real roots are returned. Set a=0 for a quadratic; set a=b=0 for a linear equation.
How it works
The calculator detects the effective degree from the leading non-zero coefficient. For a linear equation (a=0, b=0) the root is x = -d/c. For a quadratic (a=0) the discriminant is D = c squared minus 4bd; two roots come from the quadratic formula when D is positive, one repeated root when D is zero, and no real roots when D is negative. For a cubic, the substitution x = t minus b divided by 3a removes the squared term to give a depressed cubic t cubed plus pt plus q. The cubic discriminant Delta = -(4p cubed + 27q squared) determines root type. When Delta is positive, three distinct real roots are found via the trigonometric (Vieta's) method; when Delta is zero there is a repeated root; when Delta is negative there is one real root via Cardano's formula.
Worked example
Using the defaults a=1, b=-6, c=11, d=-6 gives x cubed minus 6x squared plus 11x minus 6 = 0. The cubic discriminant is positive, indicating three distinct real roots. The roots are x = 1, x = 2, and x = 3. You can verify by substitution: 1 minus 6 plus 11 minus 6 = 0; 8 minus 24 plus 22 minus 6 = 0; 27 minus 54 plus 33 minus 6 = 0. All three check out exactly.
Related calculators
- Factoring Calculator: express a quadratic as a product of two binomial factors.
- Discriminant Calculator: quickly classify the root type of a quadratic from its discriminant.
- Arithmetic Sequence Calculator: nth term and sum of an arithmetic progression.
- Percent Error Calculator: check how close your hand-calculated root is to the true value.