Polynomial Calculator

Combine Two Polynomials

Enter Polynomial A and B, choose an operation, then get the simplified expression in standard form.

Use terms like x^3, 2x^2, -5x, and constants. Parentheses and functions are not supported.

Choose an operation

The selected operation changes the transformation shown in the result.

Only one variable, x, is supported.

Polynomial result

Simplified expression Choose an operation Result and transformation steps will appear here.

What is a Polynomial?

A polynomial is an algebraic expression consisting of variables and coefficients, involving operations of addition, subtraction, multiplication, and non-negative integer exponents. Polynomials are fundamental in algebra and appear throughout mathematics and science.

Examples of polynomials include 3x² + 2x - 5 (quadratic), x³ - 4x (cubic), and 2x⁴ - x³ + 3x - 1 (quartic). The highest exponent determines the degree of the polynomial.

How Polynomial Operations Work

Follow these detailed steps:

  1. Step 1: Identify Like Terms
    Terms with the same variable and exponent can be combined. 3x² and 5x² are like terms; 3x² and 3x are not.
  2. Step 2: Apply the Operation
    For addition/subtraction: combine like terms. For multiplication: distribute each term across the other polynomial.
  3. Step 3: Simplify the Result
    Combine any like terms in the result and write in standard form (descending powers of x).

Polynomial Operations

Addition: (ax² + bx + c) + (dx² + ex + f) = (a+d)x² + (b+e)x + (c+f)

Subtraction: (ax² + bx + c) - (dx² + ex + f) = (a-d)x² + (b-e)x + (c-f)

Multiplication: (a + b)(c + d) = ac + ad + bc + bd

When multiplying polynomials, use the distributive property (FOIL method for binomials) to multiply each term systematically.

Example

Polynomial Addition Example

Problem: (2x² + 3x - 1) + (x² - 2x + 4)

Solution:

  1. Group like terms: (2x² + x²) + (3x - 2x) + (-1 + 4)
  2. Add coefficients: 3x² + x + 3
  3. Result: 3x² + x + 3

Quick Calculation Tips

  • Line up like terms vertically for addition and subtraction
  • FOIL works only for multiplying two binomials - use general distribution for more terms
  • Use standard form so the highest power appears first
  • The degree of the result equals the sum of degrees when multiplying

Common Mistakes to Avoid

  • Distributing the negative incorrectly
    Subtracting (3x - 5) means adding (-3x + 5), not (-3x - 5). The negative applies to every term.
  • Forgetting to multiply all combinations
    When multiplying (a+b)(c+d), you need ac + ad + bc + bd - all four combinations.

Frequently Asked Questions

What is the degree of a polynomial?

The degree is the highest exponent of the variable in the polynomial. For example, 3x⁴ + 2x² - 5 has degree 4.

What are like terms?

Like terms are terms that have the same variable raised to the same power. For example, 3x² and 5x² are like terms, but 3x² and 3x are not.

Can I divide polynomials?

Yes, polynomial division (long division or synthetic division) is possible but more complex. This calculator currently supports addition, subtraction, and multiplication.

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