Find roots calculator
Find all real and complex roots (zeros) of any polynomial. Shows factoring steps, rational root theorem, and verification.
What are roots of a polynomial?
The roots (or zeros) of a polynomial are the values of x where the polynomial equals zero. For y = x² − 4, the roots are x = 2 and x = −2.
Methods for finding roots
Factoring: If you can factor the polynomial, set each factor = 0.
Quadratic formula: For degree 2: x = (−b ± √(b²−4ac)) / 2a.
Rational root theorem: For degree 3+, test p/q where p divides the constant term and q divides the leading coefficient.
Synthetic division: Once you find one root, divide it out and repeat on the quotient.
Fundamental theorem of algebra
Every polynomial of degree n has exactly n roots (counting multiplicity and complex roots). A cubic always has 3 roots, a quartic always has 4, etc.
Multiplicity
If a root appears more than once, it has multiplicity > 1. For (x−2)³, x = 2 is a root with multiplicity 3. At a root with even multiplicity, the graph touches the x-axis but doesn't cross it.