Quick answer

Combine like terms only: terms with the same power of x. For addition, add coefficients; for subtraction, subtract coefficients or add −Q(x) term by term.

Formula

  • (P + Q)(x) = Σ (a_k + b_k) x^k
  • (P − Q)(x) = Σ (a_k − b_k) x^k
  • Use 0 for any missing power on either side.

Introduction

These rules are the algebra behind every worksheet row and every result line in the Adding and Subtracting Polynomials Calculator. They are short to state but easy to break in practice when signs or missing terms get ignored.

Keep this page nearby when you grade homework: if an answer adds exponents or drops a middle term, you can point to the specific rule that was violated.

What the rules require

Like terms share the same variable and the same exponent. Coefficients combine with ordinary integer or fraction arithmetic; the exponent on x never changes during add or subtract. x2 and x3 stay separate terms.

If you need a refresher on vocabulary first, read what polynomials are and how standard form lists powers. Subtraction always means change the sign on every term of Q(x), then follow the addition rule.

Alignment is part of the rule: create one slot per exponent from the maximum degree down to 0, fill missing entries with 0, then combine column by column.

Formal rules

  • Align powers from max(deg P, deg Q) down to 0.
  • Addition: new coefficient on x^k is a_k + b_k.
  • Subtraction: new coefficient on x^k is a_k − b_k (or a_k + (−b_k)).

Treat missing terms as zero coefficients so a cubic plus a linear polynomial still has a row for x2. After combining, drop terms whose coefficients became 0 unless the entire expression is 0.

For extra practice merging coefficients, work through combining like terms examples before tackling long subtraction problems. The rules here and the like-term guide describe the same operation at different speeds.

Step-by-step guide

  1. Line up powers in standard form. Write P(x) and Q(x) with highest degree first. List every exponent you need, even when one polynomial omits a power. A small table with one row per k prevents skipped columns.
  2. Add or subtract coefficients per power. Perform one arithmetic operation per exponent. For subtraction, either subtract b_k from a_k directly or rewrite Q as −Q and add. Double negatives demand extra care on constants.
  3. Simplify the result. Combine any like terms that were not aligned initially, remove zero terms, and write R(x) in standard form. If all coefficients cancel, answer 0.
  4. Check with the tool. Enter matching degrees and coefficients in polynomial calculator using Add or Subtract mode. Disagreements usually trace to a sign or a missing zero coefficient.

Example: adding two quadratics

Let P(x) = 2x2 + x and Q(x) = x2 − 3. Align columns: x2 coefficients 2 and 1 sum to 3; x coefficients 1 and 0 give 1; constants 0 and −3 give −3.

The sum is R(x) = 3x2 + x − 3. No exponent changed; only coefficients did.

Enter degree 2 with coefficients (2,1,0) and (1,0,−3) in the polynomial calculator with operation Add to confirm R(x). The same layout works when Q has more terms than P.