Leading Term
The leading term is the term with the highest degree in a polynomial, combining the leading coefficient and the variable raised to the highest power.
Definition
The leading term is the whole first term, coefficient and variable together, when a polynomial is written with the biggest exponent first; it is the leading coefficient times the variable raised to the polynomial's degree, and it controls the polynomial's long-run (end) behavior, since for large $|x|$ the polynomial "looks like" just its leading term. In a polynomial ring $F[x]$ with a monomial order, the leading term $LT(f)$ is the largest term of $f$ with respect to that order; in Grobner basis theory, the ideal of leading terms $LT(I)$ controls divisibility and reduction of polynomials in an ideal $I$, and the Buchberger algorithm builds a Grobner basis by computing S-polynomials between leading terms and reducing the remainders.
Example
In $6x^4 - 3x^2 + x - 5$, the leading term is $6x^4$; in $-2y^3 + y$, it is $-2y^3$. For $2x^5 - 9x^3 + 4x$, as $x \to +\infty$, $2x^5 \to +\infty$, and as $x \to -\infty$, $2x^5 \to -\infty$, matching the whole polynomial's behavior at the extremes. For $f = 4x^2 + 3x + 1$ under lex order, $LT(f) = 4x^2$, and dividing by $g = 2x + 5$ (where $LT(g) = 2x$) proceeds term by term using these leading terms.
Key Insight
No matter how many terms a polynomial has, for large $x$ it behaves like just its leading term, which is why end behavior depends only on degree and leading coefficient. In computational algebra, the leading term is the key handle for polynomial division and Grobner basis computation, enabling algorithmic solutions to systems of polynomial equations.