Adding and Subtracting Polynomials Calculator (combine like terms)

Understanding the polynomial addition and subtraction — concept and principle

This calculator adds or subtracts two polynomials in x by combining like terms. It tidies each input, shows the second polynomial with its signs changed when subtracting, lines up the like terms, and gives the result with its degree and a quick check at x = 1.

How it is calculated

Like terms have the same power of x. To add, add their coefficients and keep the power: 3x³ + x³ = 4x³.

To subtract, change the sign of every term of the second polynomial, then add:

(P) − (Q) = P + (−Q)

• Missing powers count as 0.
• Degree of the result = highest power with a non-zero coefficient; it can drop if the leading terms cancel.
• Check: the result at x = 1 equals the sum (or difference) of the two polynomials' coefficient sums.

The principle

A polynomial is a sum of terms, each a number times a power of x. Terms with different powers cannot be merged, just as metres and square metres cannot be added. Combining like terms is the distributive law in reverse: 3x³ + x³ = (3 + 1)x³. Subtraction distributes the minus sign over every term in the brackets, which is where most mistakes happen. Evaluating at x = 1 turns every power into 1, so it reduces each polynomial to the sum of its coefficients, a fast check.

Worked example

Default inputs: (3x³ + 2x² − 5x + 7) − (x³ − 4x² + 6).

1. Change the signs of the second polynomial: −x³ + 4x² − 6.
2. x³ terms: 3x³ − x³ = 2x³.
3. x² terms: 2x² + 4x² = 6x².
4. x terms: −5x stays −5x; constants: 7 − 6 = 1.
5. Result: 2x³ + 6x² − 5x + 1, degree 3, 4 terms. Check at x = 1: (3 + 2 − 5 + 7) − (1 − 4 + 6) = 7 − 3 = 4, and 2 + 6 − 5 + 1 = 4.

How to read the result

Write the result in descending powers, as shown. If the degree is lower than both inputs, the leading terms cancelled, which is correct, not an error. The x = 1 check catches most sign slips; a second check at x = −1 or x = 2 catches the rest.

Common mistakes

• Changing the sign of only the first term of the second polynomial.
• Adding the exponents of like terms: x² + x² is 2x², not x⁴.
• Combining unlike terms such as 2x² and 3x.
• Dropping a term whose coefficient is 1, such as x³.

Frequently asked questions

How do you subtract polynomials?

Change the sign of every term in the polynomial being subtracted, then combine like terms.

What are like terms?

Terms with the same variable raised to the same power, such as 4x² and −x².

What is the degree of a polynomial?

The highest power of x with a non-zero coefficient.

Sources: OpenStax, Elementary Algebra 2e (2020), §6.1 Add and Subtract Polynomials