Polynomial Solver
Find the roots of polynomials up to degree 4.
Quick Examples
Polynomial Roots
A root of a polynomial is a value x where f(x) = 0. The Fundamental Theorem of Algebra states that a polynomial of degree n has exactly n roots (counting multiplicity).
Quadratic Formula
Discriminant
For ax² + bx + c = 0: Δ = b² - 4ac
- Δ > 0: Two distinct real roots
- Δ = 0: One repeated real root
- Δ < 0: Two complex conjugate roots
Cubic and Quartic
Cubic equations have exact solutions (Cardano's formula). Quartic equations can be solved using Ferrari's method. Higher-degree polynomials typically require numerical methods.
What Is a Polynomial?
A polynomial is an expression built from variables and constants using only addition, subtraction, multiplication, and whole-number exponents. In standard form (highest power first):
| Term | Meaning | Example in 4x3 − 2x + 5 |
|---|---|---|
| Degree | Highest exponent present | 3 (cubic) |
| Leading coefficient | Coefficient of the highest-power term | 4 |
| Constant term | The number with no variable | 5 |
| Root (zero) | A value r with P(r) = 0 | Found by the calculator |
The calculator above handles degrees 2, 3, and 4: enter the coefficients and it returns all roots, the factored form, and the discriminant where applicable.
Finding Roots by Degree
- Degree 1 (linear): ax + b = 0 → x = −b/a. Always one root.
- Degree 2 (quadratic): Use the quadratic formula x = (−b ± √(b2 − 4ac)) / 2a. The discriminant decides: two real, one repeated, or two complex roots.
- Degree 3 (cubic): Try the Rational Root Theorem below; otherwise Cardano's formula or a numerical method finishes the job.
- Degree 4 (quartic): Ferrari's method reduces it to a cubic, but in practice the Rational Root Theorem plus the quadratic formula handles most classroom cases.
The Rational Root Theorem
For a polynomial with integer coefficients, any rational root p/q (in lowest terms) must satisfy: p divides the constant term and q divides the leading coefficient. This gives a short list of candidates to test.
Worked example: Find the roots of x3 − 6x2 + 11x − 6 = 0. Candidates: ±1, ±2, ±3, ±6 (factors of 6 over factors of 1). Test x = 1: 1 − 6 + 11 − 6 = 0 ✓ — a root. Divide out (x − 1) to get x2 − 5x + 6 = (x − 2)(x − 3). Roots: x = 1, 2, 3. Verify with Vieta's formulas: sum = 6 = −b/a ✓, pairwise sum = 11 = c/a ✓, product = 6 = −d/a ✓
Remainder and Factor Theorems
Two theorems make root-finding mechanical:
- Remainder theorem: Dividing P(x) by (x − r) leaves remainder P(r). If P(r) = 0, there is no remainder.
- Factor theorem: (x − r) is a factor of P(x) if and only if r is a root.
Synthetic division in action: Divide x3 − 6x2 + 11x − 6 by (x − 1). Bring down 1; multiply by 1 and add to −6 → −5; multiply by 1 and add to 11 → 6; multiply by 1 and add to −6 → 0. The bottom row 1, −5, 6, 0 means quotient x2 − 5x + 6 with remainder 0 — confirming x = 1 is a root.
Factoring Strategies
| Strategy | When to Use | Example |
|---|---|---|
| Greatest common factor | Every term shares a factor | 2x3 − 8x2 + 6x = 2x(x2 − 4x + 3) |
| Grouping | Four terms with pairable structure | x3 + 2x2 + 3x + 6 = (x2 + 3)(x + 2) |
| Difference of squares | a2 − b2 pattern | x4 − 16 = (x2 − 4)(x2 + 4) |
| Sum/difference of cubes | a3 ± b3 pattern | x3 − 8 = (x − 2)(x2 + 2x + 4) |
| Rational root + divide | Degree 3–4 with integer coefficients | See the worked example above |
Verify the grouping example: (x2 + 3)(x + 2) = x3 + 2x2 + 3x + 6 ✓
Real-World Applications
- Projectile motion: Height as a function of time is quadratic; with air resistance it becomes cubic.
- Engineering: Beam deflection and vibration modes are modeled with cubic and quartic polynomials.
- Economics: Cost, revenue, and profit curves are often quadratic or cubic; their roots are break-even points.
- Computer graphics: Bézier curves — the backbone of fonts and animation — are cubic polynomials.
- Signal processing: Polynomial approximations (Taylor series) underpin calculators and simulations.
Common Mistakes to Avoid
- Counting roots wrong: A degree-n polynomial has exactly n roots counting multiplicity — a quartic never has "just two" roots.
- Forgetting multiplicity: (x − 2)2 touches the x-axis at x = 2 but counts as two roots.
- Testing only positive candidates: Rational roots can be negative — always include the ±.
- Skipping the GCF: Factor out common terms first; it lowers the degree you actually have to solve.
- Assuming integer roots exist: Many polynomials have irrational or complex roots — the Rational Root Theorem finding nothing is itself information.
Multiplicity and Graph Behavior
A root's multiplicity — how many times its factor repeats — controls what the graph does at the x-axis:
- Odd multiplicity (1, 3, 5…): the graph crosses the x-axis.
- Even multiplicity (2, 4, 6…): the graph touches and turns back.
Worked example: P(x) = (x − 1)2(x + 2) has a double root at x = 1 (touch) and a single root at x = −2 (cross). Expanding: (x2 − 2x + 1)(x + 2) = x3 − 3x + 2. Check: P(1) = 1 − 3 + 2 = 0 ✓, P(−2) = −8 + 6 + 2 = 0 ✓. The factored form the calculator shows makes this behavior visible at a glance.
Descartes' Rule of Signs
Before solving, count the sign changes between consecutive nonzero terms. The number of positive real roots equals that count, minus an even number; repeat on P(−x) for negative roots.
Worked example: P(x) = x3 − 6x2 + 11x − 6 has signs +, −, +, −: three changes, so 3 or 1 positive roots. P(−x) = −x3 − 6x2 − 11x − 6 has zero changes, so no negative roots. Indeed the roots are 1, 2, 3 — three positive, zero negative ✓
Building a Polynomial from Its Roots
The process reverses too: each root r contributes a factor (x − r). To build a cubic with roots 1, 2, 3, multiply: (x − 1)(x − 2)(x − 3). Expanding stepwise: (x − 1)(x − 2) = x2 − 3x + 2, then (x2 − 3x + 2)(x − 3) = x3 − 6x2 + 11x − 6. Check the constant: (−1)(−2)(−3) = −6 ✓. Multiplying by any leading coefficient keeps the same roots.