Adding and Subtracting Polynomials Calculator

What are like terms?

Adding or subtracting polynomials comes down to one repeated skill: combining like terms. Two terms are "like terms" when they have exactly the same variable raised to exactly the same power — 3x² and −7x² are like terms; 3x² and 3x are not, even though they share a coefficient.

A polynomial is just a sum of terms, each built from a coefficient and a power of x (or the constant term, which is really x⁰, since any number to the zero power is 1). Adding two polynomials means matching up each pair of like terms — the two x⁴ terms, the two x³ terms, the two x² terms, and so on — and adding their coefficients together, one degree at a time.

The exponent never changes when combining like terms — only the coefficient does. This is the single rule underneath everything else in this article, whether the polynomials have two terms or twenty. It follows directly from the distributive property: a·xⁿ + b·xⁿ = (a + b)·xⁿ is exactly the same algebraic move as factoring xⁿ out of a sum, just run in the more familiar “combine, don’t factor” direction.

It’s worth being precise about vocabulary here, since it comes up throughout the rest of this guide. The degree of a single term is the exponent on its variable (3x⁴ has degree 4; the constant term 7 has degree 0). The degree of the whole polynomial is the highest degree among its terms. The term with that highest degree is called the leading term, and its coefficient is the leading coefficient. A polynomial with one term is a monomial, with two terms a binomial, and with three terms a trinomial — beyond three terms, it’s usually just called a polynomial with no more specific name.

Setting up the standard form

Before combining anything, both polynomials should be written in standard form: terms arranged from the highest power of x down to the constant. Missing powers are treated as having a coefficient of zero, rather than being skipped over.

For example, P(x) = 3x³ − 5x + 1 has no x² term, which is the same as saying its x² coefficient is 0. Lining P(x) up against Q(x) = x³ − 4x² + 3x + 7 term by term:

Writing out the zero placeholder explicitly, rather than leaving a gap, is what makes the term-by-term matching in the next section mechanical and mistake-resistant. This habit matters more as the polynomials get longer or messier — a five-term polynomial missing two different powers is much easier to combine correctly once every column, including the empty ones, has been written down explicitly, rather than trying to match terms by eye across two differently-formatted expressions.

Adding polynomials

To add P(x) + Q(x), add the coefficients in each column of the table above:

Working through each column one at a time, and writing the intermediate sum before simplifying, is the best defense against an arithmetic slip — especially once negative coefficients are involved, since it’s easy to drop a sign when several additions are done mentally at once.

Subtracting polynomials

Subtraction works the same way, with one extra step first: distribute the negative sign across every term of the second polynomial before combining. P(x) − Q(x) is really P(x) + (−Q(x)), and −Q(x) means flipping the sign of every single term in Q(x), not just the first one.

Once every term’s sign has been flipped, subtraction becomes ordinary addition of the modified polynomial:

Skipping the sign-flip on even one term — usually the constant, since it’s easy to treat differently from the “real” terms with an x attached — is the most common error in this entire topic. It’s covered in more depth in the common mistakes section below.

Step-by-step process

1. Write both polynomials in standard form, filling in a zero coefficient for any missing power of x.
2. For subtraction only, distribute the negative sign across every term of the second polynomial. Skip this step entirely for addition.
3. Line up like terms. Match each term in the first polynomial with the term of the same degree in the second (now sign-adjusted, if subtracting).
4. Add the coefficients within each matched pair, one degree at a time.
5. Write the final polynomial in standard form, typically omitting any terms whose combined coefficient came out to zero.

Walking through these five steps on (4x² − x + 9) − (2x² + 3x − 1) makes the process concrete. Both polynomials are already in standard form with no missing powers (step 1). Because this is subtraction, distribute the negative across the second polynomial: −(2x² + 3x − 1) = −2x² − 3x + 1 (step 2). Line up the terms: 4x² with −2x², −x with −3x, and 9 with 1 (step 3). Add each pair: 4+(−2)=2 for the x² terms, −1+(−3)=−4 for the x terms, and 9+1=10 for the constants (step 4). The final answer, already in standard form with no zero coefficients to drop, is 2x² − 4x + 10 (step 5).

Worked examples

Example 1: (2x² + 5x − 3) + (x² − 2x + 4)

Matching terms: x² terms give 2+1 = 3; x terms give 5+(−2) = 3; constants give −3+4 = 1.

Example 2: (4x³ − 2x + 6) − (x³ + 5x² − 3x + 2)

First distribute the negative across the second polynomial: −x³ − 5x² + 3x − 2. Then add term by term. There’s no x² term in the first polynomial, so treat its coefficient as 0: x³ terms give 4+(−1)=3; x² terms give 0+(−5)=−5; x terms give −2+3=1; constants give 6+(−2)=4.

Example 3: (x⁴ − x² + 1) + (−x⁴ + x² − 1)

Every pair of like terms cancels exactly: x⁴ gives 1+(−1)=0, x² gives −1+1=0, constant gives 1+(−1)=0.

This last example is a useful reminder that a nonzero-looking sum can legitimately simplify all the way down to zero — the calculator above reports this correctly rather than showing an empty or broken result.

Example 4: (−2x³ + x − 6) − (−3x³ − x² + 4)

Distribute the negative across the second polynomial first: −(−3x³ − x² + 4) = 3x³ + x² − 4. Then combine term by term, remembering the first polynomial has no x² term (coefficient 0): x³ terms give −2+3=1; x² terms give 0+1=1; x terms give 1+0=1 (the second polynomial has no x term); constants give −6+(−4)=−10.

This example is worth studying closely because it involves a negative leading coefficient in the first polynomial and a negative leading coefficient in the second, both needing careful sign-tracking through the distribution step — exactly the kind of setup where the common mistakes covered below tend to surface.

Common mistakes to avoid

Only flipping the sign of the first term when subtracting. −(x² − 3x + 5) is not −x² − 3x + 5. Every single term needs its sign flipped: −x² + 3x − 5. This is by a wide margin the most common error in this topic, and it usually happens because the negative sign only “feels” attached to the first term visually.

Forgetting that a missing power has a coefficient of zero, not that it disappears. If one polynomial has an x² term and the other doesn’t, the missing one still needs to be tracked as 0 through the combination — it isn’t simply left out of the final answer, since the other polynomial’s x² coefficient still needs to appear in the result on its own.

Combining terms that aren’t actually alike. 3x² and 3x share a coefficient but have different exponents, so they cannot be combined into a single term. Only the exponents matching makes two terms “like.”

Losing track of a sign partway through a longer polynomial. With four or five terms per polynomial, it’s easy for a minus sign to silently disappear or duplicate midway through. Writing out every intermediate step, rather than trying to combine several terms mentally at once, catches this before it reaches the final answer.

Assuming the two polynomials have their terms listed in the same order. Two polynomials aren’t always given with matching term order — one might be written 5 + 3x − x² while the other is written x² − 2x. Before combining anything, it helps to rewrite both in the same standard (highest-to-lowest degree) order, so that lining them up column by column, as in the setup section above, doesn’t accidentally pair terms of different degrees together.

Checking your answer

The fastest way to sanity-check an addition or subtraction result is to substitute a convenient value for x — usually x = 1 — into the original two polynomials and the final combined answer, and confirm the numbers agree.

Using example 1 from above, P(x) = 2x² + 5x − 3 and Q(x) = x² − 2x + 4, with result 3x² + 3x + 1:

Evaluating the answer at the same point: 3(1)² + 3(1) + 1 = 3 + 3 + 1 = 7. The two totals match, which is strong evidence (though not an absolute proof, since a coincidental error could theoretically still produce a matching value) that the combination was done correctly. Because this check takes only a few seconds, it’s worth doing as a matter of habit on every polynomial addition or subtraction problem, particularly on a test or assignment where a calculator like this one isn’t available to double-check the algebra directly.

The degree of the result

Adding or subtracting two polynomials never produces a result with a higher degree than the higher of the two inputs — combining like terms can only keep a degree’s coefficient the same, increase it, decrease it, or cancel it to zero, but it can never introduce a brand-new, higher power of x that wasn’t present in either original polynomial.

It’s also possible, as example 3 above shows, for the leading terms to cancel entirely, leaving a result of lower degree than either input — or, in the extreme case, a result of 0 altogether. This is worth watching for specifically whenever the two polynomials share the same leading term with opposite signs.

Real-world applications

Combining cost or revenue models. If a company’s fixed and variable costs are each modeled as separate polynomials in a quantity variable q, the company’s total cost is found by adding those two polynomials together, combining like terms exactly as covered throughout this article.

Physics: combining displacement functions. When a moving object’s position is described by a polynomial in time, and a second, independent motion (like wind drift, or a secondary force) is layered on top, the total position is the sum of the two polynomials — again, term by term.

Comparing two models. Subtracting one polynomial model from another — for instance, actual measured data fit to a polynomial minus a theoretical prediction polynomial — produces an “error” or “residual” polynomial whose coefficients show exactly where and how much the two models disagree at each order.

Engineering: superposition of systems. In systems described by polynomial transfer functions, combining the behavior of two independent components often means adding their describing polynomials directly, relying on the same like-term matching process used throughout this guide.

Inventory and multi-period budgeting. If a business projects revenue as one polynomial in time (say, capturing seasonal growth) and costs as a separate polynomial, net profit over time is found the same way as any other polynomial subtraction: distribute the negative across the cost polynomial’s terms, then combine like terms with the revenue polynomial. The resulting profit polynomial can then be evaluated at any specific time period just by plugging in a value for the variable.

Geometry: combining area or perimeter expressions. When a composite shape’s total area is built from several individual regions, each expressed as a polynomial in a shared variable (like a side length), the total area is the sum of those polynomials. If a smaller region needs to be excluded — a cut-out or a hole in the shape — that region’s area polynomial is subtracted instead, using exactly the sign-distribution procedure covered in this guide.