Polynomial Calculator
Evaluate, add, subtract polynomials and find derivatives
How It Works
Overview
A polynomial is an expression of the form P(x) = aₙxⁿ + aₙ₋₁x^(n−1) + ... + a₁x + a₀, where the exponents are non-negative integers and the coefficients are real numbers. Polynomials are everywhere in mathematics, science, and engineering — from curve fitting to signal processing to the equations of motion.
This calculator handles four common operations on polynomials: evaluating P(x) at a specific value of x, adding and subtracting two polynomials, and computing the derivative. Coefficients are entered from highest to lowest degree as a comma-separated list, with explicit zeros for any missing terms.
The Formula
The four operations performed by this calculator:
- Evaluate: substitute a number for x and compute the resulting value. Horner's method is used internally for efficiency.
- Add / Subtract: combine like terms — coefficients of the same degree add or subtract, others stay as-is.
- Derivative: apply the power rule term-by-term: d/dx [a·x^n] = n·a·x^(n−1). The derivative of a degree-n polynomial is a degree-(n − 1) polynomial.
By the Fundamental Theorem of Algebra, a polynomial of degree n has exactly n roots in the complex numbers (counted with multiplicity). Real-coefficient polynomials have complex roots in conjugate pairs.
Worked Example
Take P(x) = 2x³ − 5x + 4, entered as coefficients 2, 0, -5, 4 (note the 0 placeholder for the missing x² term).
- Evaluate at x = 2: P(2) = 2(8) − 5(2) + 4 = 16 − 10 + 4 = 10
- Derivative: P'(x) = 6x² − 5 (coefficients
6, 0, -5) - Add to Q(x) = x² + 3x − 1 (coefficients
0, 1, 3, -1after padding): result is 2x³ + x² − 2x + 3
Notice how addition simply combines coefficients of matching degree. The derivative drops the constant term (4) and reduces every other term's exponent by one while multiplying the coefficient by the original exponent.
When to Use This
- Algebra and pre-calculus — practicing arithmetic on polynomial expressions.
- Calculus — quickly compute derivatives of polynomial expressions to check homework or exam work.
- Curve fitting — once a regression yields a polynomial model, evaluate it at any input or differentiate to find slopes.
- Engineering models — beam deflection, drag curves, and many empirical fits are polynomials.
- Combining series expansions — Taylor and Maclaurin polynomial approximations need to be added, subtracted, or differentiated.
Common Mistakes to Avoid
- Forgetting zero placeholders. 2x³ − 5x + 4 must be entered as
2, 0, -5, 4; leaving out the 0 shifts every coefficient by one degree. - Mismatching degrees when adding or subtracting. The shorter polynomial is padded with leading zeros — make sure your alignment is correct.
- Power-rule errors on derivatives. The derivative of x⁴ is 4x³, not x³ or 4x⁴. Multiply by the original exponent and subtract one.
- Treating x⁰ as 0. Any nonzero number raised to the 0 power is 1, so the constant term's "hidden" x⁰ is just 1.
- Confusing the polynomial with its graph. A polynomial of degree n can have up to n − 1 turning points, but it might have fewer; degree alone doesn't fix the shape.
Frequently Asked Questions
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