Quick answer

Add coefficients of matching powers. Missing terms use coefficient 0 before you combine.

Formula

  • (P + Q)(x) = Σ (a_k + b_k) x^k

Introduction

Patterns below appear on homework and tests. After each, confirm with the Adding and Subtracting Polynomials Calculator in Add mode so arithmetic slips do not linger.

Read one example type at a time: monomial, binomial, then trinomial. The mechanics are identical; only the number of columns grows.

Addition in practice

Horizontal layout works for short sums: rewrite (2x + 3) + (5x − 1) and combine like terms in one line. Vertical columns help when degrees differ, because zeros keep spacing obvious.

For the full procedure, see how to add polynomials before drilling these templates.

Every example ends in standard form after combining like terms across both inputs and the sum.

Sample setups

  • Monomial + monomial: one exponent column.
  • Binomial + binomial: two or three columns depending on overlap.
  • Trinomial + trinomial: align x², x, and constant rows.

When degrees differ, pad the shorter polynomial with leading zeros. A degree-1 plus a degree-3 problem still needs an x³ row and an x² row with coefficient 0 on the shorter side.

Coefficients may be fractions; the rule unchanged: add only matching powers.

Step-by-step guide

  1. Write both polynomials in standard form. Sort descending. Simplify inside each polynomial first if terms were scrambled.
  2. Build a column for each exponent. List every power from the maximum degree down to 0. Write 0 where a polynomial skips a power.
  3. Add coefficients row by row. One arithmetic operation per column. Circle unlike terms before you start so you do not merge x and x².
  4. Check in the tool. Operation Add, matching degrees, coefficients entered high to low in polynomial calculator.

Example: trinomial sum

(x² + 2x + 3) + (2x² − x + 1). x²: 1 + 2 = 3. x: 2 + (−1) = 1. Constants: 3 + 1 = 4.

Result: 3x² + x + 4. No exponent arithmetic occurred.

Enter both trinomials as degree-2 coefficient lists in the polynomial calculator to confirm. Try a binomial plus trinomial next with an explicit zero row for the missing power.