What is a polynomial?

A polynomial is a math expression made by adding (or subtracting) terms that look like coefficient × variable^power. The variable is usually x, the coefficients are numbers, and the powers are whole numbers like 0, 1, 2, 3, … (no negative powers, no fractions in the exponent). Examples:

• Constant: 7 (this is a degree-0 polynomial)
• Linear: 2x + 5 (degree 1)
• Quadratic: x² − 4x + 3 (degree 2)
• Cubic: 3x³ + 0x² − 2x + 5 (degree 3)

The highest power that has a non-zero coefficient is called the degree. The degree matters because it tells you, at a glance, how “curvy” the function can be, how many turning points are possible, and how many roots (solutions where the polynomial equals 0) it can have.

A polynomial of degree n can be written as: P(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + … + a₂x² + a₁x + a₀, where the a values are coefficients. In this calculator, you enter those coefficients in order: aₙ, aₙ₋₁, …, a₀.

Evaluation, derivative, integral, and roots

To evaluate P(x) at a specific value (like x = 2), you plug the number into the formula. A direct approach works, but for speed and numerical stability, most calculators use Horner’s method. Horner rewrites the polynomial to reduce the number of multiplications:

Example: 3x³ + 0x² − 2x + 5 becomes ((3x + 0)x − 2)x + 5. Then you evaluate it from left to right, which is fast and less error-prone.

The derivative tells you the slope of the polynomial at any point. If P(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + … + a₁x + a₀, then: P′(x) = n·aₙxⁿ⁻¹ + (n−1)·aₙ₋₁xⁿ⁻² + … + 1·a₁.

In coefficient form, you multiply each coefficient by its power and shift the powers down by 1. That’s why derivatives are so fast to compute programmatically.

The indefinite integral is the “reverse” of the derivative. If: P(x) = aₙxⁿ + … + a₀, then: ∫P(x)dx = aₙ/(n+1) · xⁿ⁺¹ + … + a₀x + C, where C is an arbitrary constant. This calculator returns the coefficients of the integrated polynomial, plus the +C reminder.

A root is a value of x where P(x) = 0. For linear and quadratic polynomials, there are exact formulas. For higher degrees, exact solutions can be messy, so this calculator uses a practical approach: it scans across a range (like −50 to 50), looks for places where the polynomial changes sign, and then applies bisection to zoom in on the root.

This approach is great for finding real roots that cross the x-axis. However, it may miss repeated roots (where the curve just “touches” the axis) or complex roots (which don’t appear on the real line). If you suspect factoring is possible, try the Factoring Calculator as well.

Step-by-step polynomial examples

Polynomial: 3x³ − 2x + 5 → coefficients: 3, 0, -2, 5. Evaluate at x = 2.

• Using Horner: start with 3
• Multiply by 2 → 6, add 0 → 6
• Multiply by 2 → 12, add -2 → 10
• Multiply by 2 → 20, add 5 → 25

Result: P(2) = 25.

For 3x³ + 0x² − 2x + 5, the derivative is: P′(x) = 9x² + 0x − 2. Coefficients become: 9, 0, -2.

∫(3x³ − 2x + 5)dx = (3/4)x⁴ − x² + 5x + C. Coefficients are: 0.75, 0, -1, 5 (plus the constant C).

If your polynomial is x² − 4 (coefficients 1, 0, -4), the roots are x = -2 and x = 2. The scan+bisection method will detect sign changes near those points and report root estimates close to ±2.

Common mistakes (and how to avoid them)

• Forgetting zeros: If a power is missing, include a 0 coefficient (e.g., 5x⁴ + 2 becomes 5,0,0,0,2).
• Wrong order: Always enter from highest power → constant term.
• Extra spaces: Spaces are fine, but keep commas between numbers.
• Fraction input: You can enter 1/2 as 0.5 (decimals are safest).
• Roots not found: Increase the root search range or use factoring/quadratic solvers when appropriate.

If your polynomial degree is 2, you may prefer the Quadratic Equation Solver for exact roots.

Frequently Asked Questions

• Why do I have to type coefficients instead of the full expression?

  Coefficients are the most reliable input for a browser calculator without heavy parsing libraries. It avoids ambiguity like “2x^2x” or different spacing/formatting styles. If you can convert your expression into coefficients once, everything else becomes automatic and fast.

• What does “degree” mean?

  Degree is the highest power of x with a non-zero coefficient. For example, 7 is degree 0, 2x+5 is degree 1, and 3x³−2x+5 is degree 3.

• Can this calculator find complex roots?

  Not on this page. It estimates real roots by scanning for sign changes and refining with bisection. Complex roots don’t appear on the real line and require different numerical methods.

• Why might it miss a root?

  If the polynomial touches the axis and turns around (a repeated root), there may be no sign change to detect. Also, roots outside the selected range won’t be found. Try increasing the range or using factoring.

• What’s the fastest way to build coefficients?

  Write your polynomial from highest power down to constant and list each coefficient. Put 0 wherever a power is missing. Example: 4x⁵ − x³ + 9 becomes 4, 0, -1, 0, 0, 9.

Why polynomials show up everywhere

Polynomials aren’t just “school math.” They show up in curve fitting, physics, engineering, computer graphics, and even finance. A few quick examples:

• Approximation: Many functions are approximated by polynomials (Taylor series) near a point.
• Optimization: Derivatives help find maxima/minima (design, costs, trajectories).
• Data modeling: Polynomial regression is used to fit curved trends.
• Motion: Position, velocity, and acceleration can be polynomial-like in time.

In practice, being able to evaluate a polynomial fast (and derive its derivative) is a mini-superpower: it lets you sanity-check homework, verify code outputs, and debug models quickly.