Quick answer

P(x) − Q(x) means add P(x) and (−Q(x)). Flip the sign on each coefficient of Q, then combine like terms exactly as in addition.

Formula

  • (P − Q)(x) = Σ (a_k − b_k) x^k
  • Same alignment rules as addition; order matters.

Introduction

Treat subtraction as addition after negating Q(x). The Adding and Subtracting Polynomials Calculator applies that rule in Subtract mode so you can focus on layout on paper.

Parentheses around Q(x) are not optional when Q has more than one term. The minus sign must reach every coefficient inside the parentheses.

What is polynomial subtraction?

Subtraction is not a separate exponent rule. It is addition with the second polynomial negated. Watch constants and middle terms; they are the usual victims of rushed work.

Compare sign handling with polynomial addition vs subtraction when you forget whether order matters. P − Q and Q − P differ whenever Q is not zero.

After coefficients are combined, simplify R(x) the same way you would after addition.

Subtraction formula

  • Coefficient of x^k in R(x) = (coefficient of x^k in P) − (coefficient of x^k in Q).
  • Equivalent: add P(x) and (−Q(x)) term by term.
  • Use 0 for missing terms on either side.

Distribute the minus before aligning columns when Q is written in parentheses. −(x² − 2x + 1) becomes −x² + 2x − 1, not −x² − 2x − 1.

Walk through numeric cases in subtracting polynomials examples once the steps here feel familiar.

Step-by-step guide

  1. Rewrite as P(x) + (−Q(x)). Change every sign in Q(x) if that is clearer than column subtraction. Both methods are equivalent when applied completely.
  2. Align like terms. Use the same table you would for addition. Pad degrees with zero coefficients.
  3. Subtract or add negated coefficients. Careful with double negatives: subtracting a negative coefficient adds. Mark each column before moving on.
  4. Simplify and check order. Confirm the problem asked for P − Q, not Q − P. Remove zero terms and use polynomial calculator Subtract mode to verify.

Example: cubic minus cubic

Let P(x) = 2x3 + x − 1 and Q(x) = x3 + 4x − 5. Subtract columnwise: x³: 2 − 1 = 1; x²: 0 − 0 = 0; x: 1 − 4 = −3; constant: −1 − (−5) = 4.

So R(x) = x3 − 3x + 4. The zero x² term is omitted in final standard form.

Set Subtract in the tool with degree 3, enter coefficients for P and Q including zeros, and compare to your table.