Quick answer

Rewrite P − Q as P + (−Q). Flip every sign in Q, then combine like terms.

Formula

  • (P − Q)(x) = Σ (a_k − b_k) x^k

Introduction

Subtraction errors usually mean a sign never reached a term in Q(x). The examples below show correct distribution before you use polynomial calculator Subtract mode.

Work slowly through parentheses; speed without distribution is what costs points.

Sign distribution

Parentheses around Q matter when Q has more than one term. The minus applies to each addend inside: −(a + b) = −a − b.

Follow the step list in how to subtract polynomials alongside these numeric templates.

Notice how P − Q differs from Q − P in polynomial addition vs subtraction when the same expressions swap order.

Worked pattern

  • Step 1: Negate every term of Q(x).
  • Step 2: Add P(x) and −Q(x) by aligning powers.
  • Step 3: Simplify and write standard form.

Double negatives become positives: subtracting −2x is adding 2x. Mark sign changes in a different color on paper if that helps.

Binomial minus binomial problems are good drills before cubics because there are fewer columns but the same sign rules.

Step-by-step guide

  1. Parenthesize Q(x) before flipping signs. Write P − ( … all terms of Q … ). Only then distribute the minus across each term.
  2. Combine like terms. Use the addition layout on P and −Q. Pad missing powers with zero.
  3. Verify order and simplify. Confirm you computed P − Q, not Q − P. Drop zero terms unless the answer is 0.
  4. Compare with the calculator. Subtract mode in polynomial calculator should match your final R(x).

Example: quadratic subtraction

(3x² + 5) − (x² − 2). Distribute: 3x² + 5 − x² + 2. Combine x²: 3 − 1 = 2. Constants: 5 + 2 = 7.

Result: 2x² + 7. The x terms were absent (coefficient 0) on both sides.

Enter coefficients for both quadratics in polynomial calculator Subtract mode to verify.