Add & Subtract Polynomials Calculator

Enter polynomial coefficients for x4 through x0, choose addition or subtraction, and see the step-by-step result with proper formatting.

Enter Polynomial Coefficients

Polynomial 1: P(x)

P(x) = 2x2 + 3x + 1

Polynomial 2: Q(x)

Q(x) = x2 - 2x + 4

Result

Result Polynomial
3x2 + x + 5
Degree 2
Number of Terms 3
Classification Trinomial

Step-by-Step: Combining Like Terms

(2x^2 + 3x + 1) + (x^2 - 2x + 4) = 3x^2 + x + 5

What Are Polynomials?

A polynomial is an algebraic expression consisting of variables (usually x), coefficients (numbers), and non-negative integer exponents, combined using addition, subtraction, and multiplication. For example, 3x2 + 2x - 5 is a polynomial of degree 2.

Polynomial Terminology

  • Term: Each part separated by + or - signs (e.g., 3x2 is a term).
  • Coefficient: The numerical factor in a term (e.g., 3 in 3x2).
  • Degree: The highest exponent in the polynomial. The degree of 3x2 + 2x - 5 is 2.
  • Leading coefficient: The coefficient of the highest-degree term.
  • Constant term: The term with no variable (x0 term).

Like Terms

Like terms are terms that have the same variable raised to the same power. For example, 3x2 and 5x2 are like terms because both contain x2. However, 3x2 and 3x3 are NOT like terms because the exponents differ. When adding or subtracting polynomials, you can only combine like terms.

How to Add Polynomials

  1. Write both polynomials in standard form (descending order of exponents).
  2. Align like terms (same exponent).
  3. Add the coefficients of like terms.
  4. Write the result in standard form.

How to Subtract Polynomials

  1. Distribute the negative sign to the second polynomial (change all signs).
  2. Then add the resulting polynomials as described above.

Polynomial Classification

Monomial

A polynomial with exactly one term.

Example: 5x3

Binomial

A polynomial with exactly two terms.

Example: x2 + 3

Trinomial

A polynomial with exactly three terms.

Example: x2 + 2x + 1

Degree Classification

Constant (0), Linear (1), Quadratic (2), Cubic (3), Quartic (4).

Degree = highest exponent

Important Properties

  • Adding or subtracting polynomials always results in another polynomial.
  • The degree of the result is at most the maximum degree of the two polynomials (it can be less if leading terms cancel).
  • Only like terms (same variable, same exponent) can be combined.
  • The commutative and associative properties apply to polynomial addition.