Polynomial Calculator

Evaluate polynomials, add or multiply polynomials, and find roots by substitution. Enter coefficients for polynomials up to degree 4 and compute values at any x.

∑ Algebra📐 p(x) = ax⁴ + bx³ + cx² + dx + e🔢 Math

Coefficient a (x⁴)

Coefficient b (x³)

Coefficient c (x²)

Coefficient d (x)

Constant e

Evaluate at x =

Please enter valid values.

Formula & Reference

Variable	Symbol	Formula	Units
Polynomial Calculator	—	p(x) = ax⁴ + bx³ + cx² + dx + e	numeric value

Step-by-Step Examples

Example 1

Cubic Evaluation

p(x) = x³ − 3x² + 2x + 5. Evaluate at x=2.

p(2) = 8 − 12 + 4 + 5 = 5

✓ p(2) = 5

Example 2

Root Check

Is x=1 a root of x³ − 3x + 2?

p(1) = 1 − 3 + 2 = 0
Yes! x=1 is a root

✓ x=1 is a root

Example 3

Quadratic

p(x) = 2x² − 5x + 3. At x=3.

p(3) = 18 − 15 + 3 = 6

✓ p(3) = 6

Real-World Applications

📚

Algebra

Core topic in every algebra course — evaluating and analyzing polynomials.

📊

Curve Fitting

Polynomial regression fits data with polynomial curves.

🧪

Engineering

Transfer functions in control systems are ratios of polynomials.

💻

Computer Science

Polynomials used in hash functions, error correction codes, cryptography.

Common Mistakes to Avoid

⚠️

Forgetting exponent order

Standard form: highest degree first. x³ + x² + x + constant.

⚠️

Mixing up coefficients

In ax² + bx + c, a is for x², b for x, c is the constant. Don't assign them to wrong powers.

⚠️

Root vs factor

x=r is a root means p(r)=0. It means (x−r) is a factor. These are related but distinct statements.

Frequently Asked Questions

What is a polynomial? ▾

An expression of the form aₙxⁿ + ... + a₁x + a₀. The degree is the highest power with nonzero coefficient.

What is the Factor Theorem? ▾

(x−r) is a factor of p(x) if and only if p(r)=0. Used to find roots and factor polynomials.

What is the Remainder Theorem? ▾

When p(x) is divided by (x−r), the remainder equals p(r). Faster than long division to evaluate.

How many roots does a polynomial have? ▾

A degree-n polynomial has exactly n roots (counting complex and repeated roots), by the Fundamental Theorem of Algebra.

What is synthetic division? ▾

A shorthand for dividing a polynomial by (x−r). Faster than polynomial long division for this special case.

Related Math Calculators

Formula Explorer connections

Interpretation: This formula describes an algebraic relationship among variables, functions, equations, roots or sequences. Assumption: Respect the expression’s domain and excluded values. Check roots in the original equation because transformations can introduce extraneous solutions.

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