Adding and Subtracting Polynomials Calculator
Enter the coefficients of two polynomials (up to degree 6 each), choose addition or subtraction, and get the fully simplified result. The step-by-step panel shows how like terms are identified and combined, so you can follow every move.
How to add and subtract polynomials
To add two polynomials, group all terms that share the same variable power (called like terms), then add their coefficients. The variable part stays unchanged. For example, 3x^2 and -5x^2 are like terms because both involve x raised to the second power; their sum is (3 + (-5))x^2 = -2x^2. Terms with different powers, such as 3x^2 and 2x, cannot be combined. To subtract Q(x) from P(x), first distribute the minus sign through every term of Q(x), flipping each sign, then proceed with addition exactly as before. The degree of the result is at most the larger of the two input degrees, and can be lower if the leading terms cancel.
Using this calculator
Choose whether you want addition or subtraction at the top, then set the degree for each polynomial using the degree selectors. Coefficient input fields appear for each power up to your chosen degree; enter 0 for any power you do not want. The result polynomial, its degree, and its type update instantly. The "Show your work" panel walks through every step: writing the operation, distributing the minus sign if needed, identifying each group of like terms, and writing the final simplified polynomial.
Why the result degree can be lower than expected
When you add or subtract two polynomials of the same degree, the leading coefficients might cancel. For instance, (4x^3 + 2x) - (4x^3 - x + 1) = 3x - 1, a degree-1 result even though both inputs were degree 3. This is called cancellation of the leading term, and it is an important thing to check when you need to know the degree of the answer in advance. This calculator shows you the actual degree of the simplified result, not just the maximum of the two input degrees.
Common mistakes to avoid
The most frequent error is combining unlike terms, for example adding 3x^2 and 2x to get 5x^3 instead of leaving them as separate terms. A second common error is forgetting to distribute the minus sign when subtracting: (a - b)x^2 - (c + d)x becomes a - b)x^2 - cx - dx, so every term of the second polynomial changes sign. A third mistake is dropping a term when it looks like it vanishes but actually equals zero, such as assuming 5x - 5x disappears with no trace; it does give zero, but it is still a valid step to record. This calculator shows each like-terms group explicitly so you can catch and correct those errors.
Polynomial types and terminology
| Name | Number of terms | Example |
|---|---|---|
| Monomial | 1 | 5x^3 |
| Binomial | 2 | 3x^2 - 7 |
| Trinomial | 3 | x^2 + 2x - 1 |
| Polynomial | 4 or more | x^4 - 3x^3 + x^2 - x + 9 |
Standard classifications used in algebra for polynomials in one variable.
Frequently asked questions
What are like terms in a polynomial?
Like terms are terms that have identical variable parts, meaning the same variable raised to the same power. For example, 7x^3 and -2x^3 are like terms (both x cubed), but 7x^3 and 7x^2 are not. You can only add or subtract the coefficients of like terms; the variable part never changes when combining them.
Does the order of terms in the result matter?
By convention, polynomials are written in descending order of degree (highest power first), which is what this calculator does. The order does not change the mathematical value, but descending order makes it easiest to read the degree and compare polynomials at a glance.
What happens when the leading terms cancel?
If the coefficients of the highest-degree terms are equal and opposite (one positive, one negative), they sum to zero and disappear from the result. The degree of the result then equals the next highest power that does not cancel. This calculator reports the actual degree of the simplified answer, not the maximum degree of the inputs.
Can I use this calculator for polynomials with more than one variable?
This tool handles polynomials in one variable (x) up to degree 6. For expressions with two or more variables, such as 3x^2y - 4xy^2, you need a multi-variable symbolic algebra tool. The single-variable restriction makes the step-by-step working much clearer for learning purposes.
How do I enter a polynomial like 2x^3 - 5 (missing the x^2 and x terms)?
Set the degree to 3, enter 2 for the coefficient of x^3, enter 0 for x^2, enter 0 for x, and enter -5 for the constant term. The calculator treats a coefficient of 0 as an absent term and will not display it in the result.
What is the difference between a binomial and a trinomial?
A binomial has exactly two non-zero terms, for example 3x^2 - 7. A trinomial has exactly three, for example x^2 + 4x - 9. A polynomial with only one term is called a monomial, and one with four or more terms is simply called a polynomial (or sometimes a multinomial). This calculator labels the result with the correct term count.