How the Adding and Subtracting Polynomials Calculator works
Adding and subtracting polynomials both come down to one rule: combine like terms. A polynomial in x is a sum of terms of the form c·x^n, where c is a coefficient and n is a non-negative whole-number exponent. Two terms are "like terms" only when they share the exact same exponent — 3x^2 and -8x^2 combine, but 3x^2 and 3x do not.
Formula and method
For polynomials A(x) = sum of a_n·x^n and B(x) = sum of b_n·x^n, addition and subtraction combine coefficients power by power:
- Addition: (A + B)(x) = sum of (a_n + b_n)·x^n
- Subtraction: (A - B)(x) = sum of (a_n - b_n)·x^n
In practice, subtraction is addition of the negative: distribute the minus sign across every term of B first (flip each of its coefficients), then add like terms exactly as you would for addition. For example, (3x^2 - 5x + 7) - (x^2 + 2x - 4) becomes 3x^2 - 5x + 7 - x^2 - 2x + 4, which simplifies to 2x^2 - 7x + 11. Terms with no match in the other polynomial carry through unchanged; a matching pair whose coefficients cancel drops out of the result entirely, which is why the result's degree can be lower than either input's degree.
Common sources of error
- Sign errors on subtraction: forgetting to flip the sign of every term in the second polynomial, not just the first one, is the most common mistake.
- Combining unlike terms: x^2 and x are different powers and can never be merged into one term, even though both contain x.
- Losing a term to a coefficient of zero: if two like terms cancel exactly, that power vanishes from the answer — it is correct, not a missing step.
Checking your result
Two quick checks catch most mistakes. First, the degree of a sum or difference can never exceed the higher of the two input degrees (it can only drop, if leading terms cancel). Second, pick an easy value of x, such as x = 1, and confirm that evaluating the result matches A(1) combined with B(1) computed separately; if the numbers disagree, a term was combined incorrectly.
Applications
Combining polynomials this way is the algebra behind simplifying expressions, adding functions in calculus (f(x) + g(x)), expressing combined perimeters or areas algebraically, and merging polynomial models in physics and engineering. Always report the result in standard form (descending powers of x) so it can be read, compared, or plugged into the next step without ambiguity.