Subtracting Polynomials
Subtracting polynomials means distributing the negative sign across the second polynomial, then combining like terms.
Definition
Subtracting polynomials means distributing the negative sign across every term of the second polynomial (rewriting $P - Q$ as $P + (-Q)$, where $-Q$ negates every coefficient in $Q$), then combining like terms as in addition. Polynomial subtraction is the additive-inverse operation in the ring $F[x]$: since $F[x]$ is an abelian group under addition, subtraction is always defined and the result is always a polynomial.
Example
$(5x^2 + 3x) - (2x^2 - x + 4) = 5x^2 + 3x - 2x^2 + x - 4 = 3x^2 + 4x - 4$. $(3x^3 - 2x + 7) - (x^3 + 4x^2 - x - 3) = 2x^3 - 4x^2 - x + 10$. In $\mathbb{Z}_5[x]$ (polynomials mod $5$), $(4x^2 + 3) - (2x^2 + 4) = 2x^2 - 1 = 2x^2 + 4$, since $-1 = 4 \bmod 5$.
Key Insight
A common error is forgetting to distribute the negative sign to every term of the second polynomial, especially the constant, so always rewrite subtraction as addition of the negated polynomial first. Over modular coefficient rings, subtraction stays well-defined but the arithmetic wraps around, a fact connected to coding theory, where polynomials over finite fields build error-correcting codes.