Quick answer

To add P(x) and Q(x), add the coefficients of each power of x. If a power appears in only one polynomial, treat the missing coefficient as 0.

Formula

  • (P + Q)(x) = Σ (a_k + b_k) x^k
  • Degree of R(x) is at most the larger of deg P and deg Q.

Introduction

When both expressions use the same variable, addition is careful bookkeeping. The Adding and Subtracting Polynomials Calculator shows P(x), Q(x), and R(x) side by side so you can verify homework quickly.

Start by writing each polynomial in descending power order. That single habit prevents most skipped middle terms before you touch any coefficients.

What is polynomial addition?

Polynomial addition adds two functions term by term. You never add exponents; you only add coefficients when the exponents already match. The result is another polynomial.

Before adding, many students simplify each input by combining like terms inside P or Q. Addition then merges matching powers from the two polynomials.

The degree of the sum cannot exceed the larger input degree, though it may drop if leading coefficients cancel.

Addition formula

  • If P(x) = a_n x^n + … + a_0 and Q(x) = b_m x^m + … + b_0,
  • then (P + Q)(x) uses coefficient (a_k + b_k) on each x^k.
  • Pad with zeros so both polynomials share the same top power.

Vertical layout helps when degrees differ: one column per power, coefficients in rows for P and Q, sum row for R. Horizontal layout works for short binomials.

After you master the steps here, study adding polynomials examples for monomial-through-trinomial patterns that appear on tests.

Step-by-step guide

  1. Write P(x) and Q(x) in standard form. Highest power first, constant last. Combine like terms within each polynomial if the problem gave unsimplified inputs.
  2. Align like terms. Each row or column represents the same power of x. Insert 0 for missing powers on the shorter polynomial so every exponent through max(n,m) appears.
  3. Add coefficients column by column. Use integer rules, including negatives. One sum per exponent; do not merge x and x².
  4. Write R(x) and simplify. Drop terms with coefficient 0 unless the entire answer is 0. State R(x) in standard form and compare with polynomial calculator Add mode.

Example: trinomial plus trinomial

Let P(x) = 3x2 + 2x + 5 and Q(x) = x2 − 4x + 1. Column for x²: 3 + 1 = 4. Column for x: 2 + (−4) = −2. Constants: 5 + 1 = 6.

Therefore R(x) = 4x2 − 2x + 6. Every step changed coefficients only.

Enter degree 2 and coefficients (3,2,5) and (1,−4,1) in the polynomial calculator with operation Add to confirm. Try reordering P and Q on paper to see addition is commutative.