Subtracting Polynomials
AlgebraSubtracting polynomials means distributing the negative sign across the second polynomial, then combining like terms.
Definition
Subtracting polynomials means changing the signs of all terms in the second polynomial (distributing the minus sign), then adding like terms.
Example
$(5x^2 + 3x) - (2x^2 - x + 4) = 5x^2 + 3x - 2x^2 + x - 4 = 3x^2 + 4x - 4$.
Key Insight
Distribute the negative sign to every term in the second polynomial before combining. The minus sign changes every $+$ to $-$ and every $-$ to $+$ in the second polynomial.
Definition
To subtract polynomial $Q$ from polynomial $P$, rewrite $P - Q$ as $P + (-Q)$, where $-Q$ is obtained by negating every coefficient in $Q$. Then combine like terms as in addition. This is equivalent to distributing a $-1$ factor across all terms of $Q$.
Example
$(3x^3 - 2x + 7) - (x^3 + 4x^2 - x - 3) = 3x^3 - 2x + 7 - x^3 - 4x^2 + x + 3 = 2x^3 - 4x^2 - x + 10$.
Key Insight
A common error is forgetting to distribute the negative sign to all terms in the second polynomial, especially the constant term. Always rewrite as addition of the negated polynomial first.
Definition
Polynomial subtraction is the additive inverse operation in the ring $F[x]$. Subtracting $f - g$ is equivalent to adding $f + (-g)$ where $-g$ is the additive inverse (all coefficients negated). The polynomial ring $F[x]$ is an abelian group under addition, so subtraction is always defined and the result is always a polynomial.
Example
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
Polynomial subtraction over modular coefficient rings behaves differently from over $\mathbb{R}$: it is always well-defined but the arithmetic is modular. This connects to coding theory, where polynomials over finite fields are used for error-correcting codes.