Calculate roots of polynomials with step-by-step solutions. Perfect for quadratic, cubic, and higher-degree equations in algebra and mathematics.
Enter coefficients from highest to lowest degree
Comprehensive polynomial root calculations with detailed explanations
Calculate quadratic equation roots
Calculate cubic equation roots
Calculate quartic equation roots
Find rational roots using theorem
Understanding polynomial equations and root-finding methods
Input coefficients of your polynomial equation.
Use appropriate root-finding algorithms for the equation type.
Receive roots, factorization, and step-by-step solutions.
x = (-b ± √(b² - 4ac)) / 2a
For cubic equations
For quartic equations
p/q form testing
Common polynomial equations and their roots
Result: x = 2, x = 3
Result: x = 1, x = 2, x = 3
Result: Time of flight calculation
Result: Critical point analysis
Common questions about polynomial roots calculations
Polynomial roots are the values of x that make the polynomial equal to zero. For example, the roots of x² - 5x + 6 = 0 are x = 2 and x = 3.
Real roots are real numbers that satisfy the equation. Complex roots involve imaginary numbers (i = √(-1)) and often come in conjugate pairs for polynomials with real coefficients.
Use the quadratic formula: x = (-b ± √(b² - 4ac)) / 2a. The discriminant (b² - 4ac) determines the nature of roots: positive = two real roots, zero = one repeated root, negative = two complex roots.
The rational root theorem states that any rational root p/q of a polynomial must have p as a factor of the constant term and q as a factor of the leading coefficient.
Cubic equations can be solved using Cardano's method, which transforms the equation into a depressed cubic form. For practical purposes, rational root theorem is often used first to find one root, then polynomial division reduces it to a quadratic.
Polynomial factorization expresses a polynomial as a product of lower-degree polynomials. For example, x² - 5x + 6 = (x - 2)(x - 3), where the roots are the values that make each factor zero.
All polynomials can be factored over the complex numbers, but not all can be factored over the rational or real numbers. The Fundamental Theorem of Algebra guarantees that every polynomial has exactly n roots (counting multiplicity) over the complex numbers.
Multiplicity refers to how many times a root appears. For example, if (x - 2)² appears in the factorization, then x = 2 is a root with multiplicity 2. The sum of all multiplicities equals the degree of the polynomial.
Substitute each root back into the original polynomial equation. If the result is zero, the root is correct. You can also use polynomial factorization to verify that the roots correspond to the factors.
Root finding is used in physics (projectile motion, oscillations), engineering (optimization, control systems), economics (break-even analysis), computer graphics (intersections), and many other fields where polynomial equations model real-world phenomena.
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