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Adding or subtracting polynomials sounds more intimidating than it actually is: the entire operation boils down to combining "like terms" — matching up terms that share the same power of x and adding or subtracting their coefficients, exactly the way you'd combine "3 apples + 2 apples" into "5 apples" without touching any oranges mixed in. A term like 3x² can only combine with another x² term, never with an x term or a plain constant, since they represent fundamentally different quantities. Where students most often stumble isn't the addition itself, but subtraction: subtracting an entire polynomial means distributing a negative sign across every single term inside the second polynomial's parentheses, not just the first one — a step that's easy to do carelessly by hand, especially with longer polynomials. This tool adds or subtracts two polynomials by combining their like terms automatically, for polynomials up to degree 4, showing every step of the process.
How it works
Each polynomial is entered as a list of coefficients, one for each power of x from the highest degree down to the constant term. For addition, coefficients of matching powers are simply added together. For subtraction, the sign of every single term in the second polynomial is flipped first — since subtracting an entire expression is mathematically equivalent to adding its opposite — and only then are the coefficients of matching powers combined. If the two polynomials have different degrees, any missing higher-degree terms in the shorter one are treated as having a coefficient of 0, so the alignment always works correctly.
- Enter operation.
- Enter p₁: x⁴ coefficient (optional).
- Enter p₁: x³ coefficient (optional).
- Enter p₁: x² coefficient.
- Enter p₁: x coefficient.
- Enter p₁: constant.
- Enter p₂: x⁴ coefficient (optional).
- Enter p₂: x³ coefficient (optional).
- Enter p₂: x² coefficient.
- Enter p₂: x coefficient.
- Enter p₂: constant.
- Click Calculate to see your results.
Examples
Adding two polynomials
(3x²+2x-1) + (x²-4x+5) = 4x²-2x+4.
Subtracting two polynomials
(3x²+2x-1) - (x²-4x+5) = 2x²+6x-6.
Adding polynomials of different degrees
(2x³+5x-3) + (4x²-x+7) treats the missing x³ term in the second polynomial and the missing x² term in the first as having a coefficient of 0, giving 2x³+4x²+4x+4 as the combined result.
Who should use it
- Algebra coursework on polynomial operations.
- Quickly checking polynomial arithmetic homework.
- Combining polynomial expressions as part of a larger algebra problem.
Industry applications
- Mathematics education
Advantages
- Supports polynomials up to degree 4.
- Shows the like-term combination clearly.
- Automatically handles polynomials of different degrees.
Limitations
- Coefficients only — does not parse a typed expression like "3x^2+2x-1".
Common mistakes to avoid
- Only flipping the sign of the first term when subtracting, instead of every term in the second polynomial.
- Trying to combine terms with different powers of x, such as adding a x² term directly to an x term.
- Misaligning terms when a polynomial is missing a middle-degree term, instead of treating it as having a coefficient of 0.
Best practices
- Line up terms by matching power of x before combining, especially when a polynomial is missing a middle term.
- When subtracting, write out the sign-flipped version of the second polynomial as an explicit intermediate step, rather than trying to distribute the negative sign and combine terms in your head at once.
- Double check your final answer by confirming every power of x in the result came from a valid combination of like terms.
Tips
- Write out both polynomials with every power of x present (using a 0 coefficient for missing terms) to avoid mis-aligning terms by hand.
- When subtracting, distribute the negative sign across the entire second polynomial before combining any terms.