Add Subtract Polynomials Calculator

Background.

Adding two polynomials is bookkeeping: terms with the same power of x combine, and nothing else interacts. In (3x² + 2x + 1) + (5x + 4) there is no x² term on the right to pair with, so 3x² carries over unchanged, while 2x + 5x gives 7x and 1 + 4 gives 5. The answer is 3x² + 7x + 5.

Hand-work mistakes usually come from misaligned powers — an x² coefficient added to an x coefficient because they sat in the same column — or from subtraction. A − B means subtracting every term of B, so the minus sign has to reach each coefficient, and a negative coefficient in B turns into an addition. (x² + 2x − 1) − (x² − 3x + 4) is 5x − 5; forget to carry the sign to the constant and you get 5x + 3, forget it on the x term and you get −x − 5.

Here each polynomial is entered as a list of coefficients starting with the constant, so 1 + 2x + 3x² is typed as 1, 2, 3, and 4 + 5x as 4, 5. Written that way, alignment is automatic: first numbers combine with first numbers, second with second, and a shorter list is padded with zeros. A missing power still needs an explicit 0 — x³ + 1 is 1, 0, 0, 1 — because a skipped position would shift every later coefficient down a power.

The combined polynomial is then evaluated at an x of your choosing, which doubles as a check on hand work: (A + B)(x) must equal A(x) + B(x). Two quirks are worth knowing. Each list needs at least two numbers, so a constant such as 7 goes in as 7, 0. And the result is not tidied, so if subtraction cancels the highest power, its 0 stays at the end of the list.

What is add subtract polynomials calculator?

The sum or difference of two polynomials is the polynomial whose coefficient for each power of x is the sum or difference of the two original coefficients for that power. Its degree is at most the larger of the two degrees, and it can be lower when subtraction cancels the leading terms. Evaluating it at a number x gives the same value as evaluating each polynomial separately and then adding or subtracting.

How to use this calculator.

1. 01Write polynomial A in ascending powers and type its coefficients in that order, separated by commas or spaces: 1 + 2x + 3x² becomes 1, 2, 3. Put a 0 in place of any missing power.
2. 02Enter polynomial B the same way. It may be shorter or longer than A; the shorter list is treated as having zeros for its missing higher powers.
3. 03Choose Add, or Subtract B from A. For B − A, swap the two entries.
4. 04Enter the x at which to evaluate the result — 1, 2 or 10 make easy mental checks — then read the value and the result's coefficients, constant first.

The formula.

If A = a₀ + a₁x + a₂x² + … and B = b₀ + b₁x + b₂x² + …, then A ± B has coefficients c_k = a_k ± b_k for every power k, with a missing a_k or b_k taken as 0 past the end of the shorter list. This is the distributive law applied to like terms: a_k x^k + b_k x^k = (a_k + b_k) x^k, while terms with different powers never merge. The result is then evaluated as P(x) = c₀ + c₁x + c₂x² + …, each coefficient multiplied by x raised to its position in the list, counting from zero. Coefficients may be negative or decimal. Evaluation is the natural place to catch a typing slip: the value at any x must equal A(x) ± B(x), so if you work those two out separately by hand, the three numbers must agree. Choosing x = 1 is especially quick, since P(1) is just the sum of the coefficients.

A worked example.

Frequently asked questions.

References& sources.

1. [1]OpenStax, Rice University. College Algebra 2e, 2021. Algebra and function chapters. Retrieved 2026-08-06.
2. [2]OpenStax, Rice University. Precalculus 2e, 2021. Trigonometry and analytic geometry chapters. Retrieved 2026-08-06.
3. [3]NIST/SEMATECH. e-Handbook of Statistical Methods, 2012. Chapter 1.3 and Chapter 4. Retrieved 2026-08-06.