Polynomial Addition & Subtraction Calculator
Add or subtract two polynomials by combining like terms, with the negative sign distributed step by step for subtraction, a column-by-column visual showing exactly how each power group combines, a numeric spot-check that plugs in a real number to confirm the result, and automatic classification by degree and term count.
Background
A polynomial is a sum of terms, where each term is a coefficient multiplied by a variable raised to a whole-number power. To add two polynomials, combine the coefficients of matching like terms (same power) and leave everything else untouched. To subtract one polynomial from another, first distribute the negative sign across every term of the second polynomial — flipping each of its signs — then combine like terms exactly as in addition. This calculator handles any single variable letter, missing terms, negative coefficients, decimals, and even messy double signs like "−−5", and shows the full breakdown.
How to use this calculator
- Choose Add or Subtract for how the two polynomials should be combined.
- Type each polynomial using ^ for powers, e.g. 3x^2 + 5x - 7. Any single letter works as the variable, and terms can be missing (a "gap") or have decimal/negative coefficients.
- Click Combine to see the result, the column-by-column breakdown, and (for subtraction) exactly how the negative sign gets distributed.
- Turn on the numeric spot-check to see the whole calculation verified by plugging in an actual number for the variable — a quick, concrete way to sanity-check any symbolic result.
- Turn on degree & classification details to see how the result is named (linear, quadratic, trinomial, etc.), a side-by-side degree comparison, and whether its degree dropped due to cancellation.
How adding and subtracting polynomials works
A polynomial is a sum of terms, each a coefficient times the variable raised to a whole-number power. "Like terms" are terms that share the exact same power — only like terms can ever combine.
Adding two polynomials means adding the coefficients of matching like terms and leaving every other term unchanged — nothing multiplies, and no exponent ever changes.
Subtracting a polynomial is identical to adding its opposite: distribute the negative sign across every single term of the second polynomial first, flipping each of its signs, then combine like terms exactly as in addition.
Terms with different powers never combine into one term — a lone x² term and a lone x term remain two separate terms in the result no matter what.
If two like terms are exact opposites (like 6x² and −6x²), they cancel completely and vanish from the result — this can even lower the overall degree of the answer below both starting degrees, or leave a "gap" of missing powers in the middle of the result.
Results are written in standard form: exponents listed from highest to lowest, with any exponent that ended up with a zero coefficient simply left out. A double sign like "−(−5)" always simplifies to a single sign first (two negatives make a positive) before combining.
Because combining like terms never changes the polynomial's value — only how it's written — substituting any number for the variable into the original expression and into the simplified result must always produce the exact same number. That's a fast way to check your work by hand.
Formulas & Equations Used
Addition: (Σ aᵢxᵢ) + (Σ bᵢxᵢ) = Σ (aᵢ+bᵢ)xᵢ
Subtraction: (Σ aᵢxᵢ) − (Σ bᵢxᵢ) = Σ (aᵢ−bᵢ)xᵢ, found by first negating every bᵢ
Degree: the largest exponent whose coefficient is not zero, after combining
Standard form: terms written with exponents in strictly decreasing order
Numeric spot-check: for any number k, A(k) ± B(k) = Result(k) must hold exactly, since combining like terms never changes the value of the expression
Example Problems & Step-by-Step Solutions
Example 1 — Basic addition
(3x² + 5x − 7) + (2x² − 3x + 4)
Step: x²: 3+2=5. x: 5+(−3)=2. const: −7+4=−3.
Result: 5x² + 2x − 3 — a quadratic trinomial.
Example 2 — Subtraction with sign distribution
(5x³ − 2x² + 4x − 7) − (2x³ + 3x² − x + 5)
Step: x³: 5−2=3. x²: −2−3=−5. x: 4−(−1)=5. const: −7−5=−12.
Result: 3x³ − 5x² + 5x − 12 — a cubic polynomial.
Example 3 — Missing terms (a "gap" in the input)
(4x³ − 5) + (2x² + 3x)
Step: Treat missing powers as coefficient 0: x³: 4+0=4. x²: 0+2=2. x: 0+3=3. const: −5+0=−5.
Result: 4x³ + 2x² + 3x − 5 — every gap gets filled in by the other polynomial's term.
Example 4 — Opposite polynomials cancel completely
(6x² − 4x + 9) − (6x² − 4x + 9)
Step: Every like term is identical, so every column subtracts to 0: x²: 6−6=0, x: −4−(−4)=0, const: 9−9=0.
Result: 0 — the zero polynomial. Subtracting a polynomial from itself always gives 0.
Example 5 — Leading terms cancel, degree drops
(x⁴ − 3x³ + 2x − 1) + (−x⁴ + x³ − x² + 5)
Step: x⁴: 1+(−1)=0 (cancels!). x³: −3+1=−2. x²: 0+(−1)=−1. x: 2+0=2. const: −1+5=4.
Result: −2x³ − x² + 2x + 4 — the degree drops from 4 to 3 because the leading terms were opposites.
Example 6 — A sparse result (a gap in the middle)
(5x⁴ + 3x² − 7) − (3x² − 2)
Step: x⁴: 5−0=5. x²: 3−3=0 (cancels!). const: −7−(−2)=−5.
Result: 5x⁴ − 5 — the x³, x², and x terms are all missing; the leading term survives but the middle vanishes entirely.
Example 7 — A different variable letter
(3t² + 2t − 5) − (t² − 4t + 1)
Step: t²: 3−1=2. t: 2−(−4)=6. const: −5−1=−6.
Result: 2t² + 6t − 6 — the same rules work for any single variable letter.
Example 8 — Messy double signs
3x² − −5 (a typo-like double negative)
Step: Two negatives in a row simplify to one positive first: 3x² − −5 becomes 3x² + 5.
Result: 3x² + 5 — the calculator normalizes stray double signs like this automatically before combining.
Frequently Asked Questions
What counts as "like terms"?
Terms that share the exact same variable raised to the exact same power. Coefficients can be anything, but the exponent has to match exactly — 3x² and 5x² are like terms, but 3x² and 5x are not.
Why do I distribute a negative sign when subtracting?
Subtracting a polynomial is the same as adding its opposite. Flipping the sign of every one of its terms first turns the subtraction into a plain addition of like terms, which is why that distribution step comes first.
Can the result have a lower degree than either original polynomial?
Yes. Whenever the highest-power terms of the two polynomials are exact opposites, they cancel out, and the degree of the answer drops below the higher of the two starting degrees.
What if a term is missing from one polynomial?
Treat the missing power as having a coefficient of 0. It still lines up in that column, it just doesn't add or subtract anything — the other polynomial's coefficient for that power carries straight through.
Can the two polynomials use different letters, like x and y?
No — this calculator combines single-variable polynomials, so both inputs need to use the same variable letter (plain numbers are always fine on their own).
How is a polynomial classified?
By term count — monomial (1), binomial (2), trinomial (3) — and by degree, the highest exponent present: linear (1), quadratic (2), cubic (3), quartic (4), quintic (5), and "degree n" beyond that.
What does the numeric spot-check actually prove?
It plugs a real number into both original polynomials and into the simplified result, then checks that the arithmetic matches. It's not a full proof, but if the numbers ever disagree, you know a mistake was made somewhere in combining like terms — it's the same "pick a number and check" trick many textbooks recommend.
What happens if I type something like "3x^2 − −5" by mistake?
The calculator treats a run of repeated signs the way algebra does: an even number of minus signs in a row becomes a plus, and an odd number stays a minus. So "− −5" becomes "+5" automatically instead of producing a wrong answer.
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