leading coefficient
The leading coefficient is the number in front of the highest-power term. Once you line a polynomial up in descending order, the term out front is the leading term, and its coefficient is the leading coefficient. In 3x^2 - 5x + 7, that number is 3.
This number is more important than it looks. It controls the overall shape and direction of the graph for large inputs: a positive leading coefficient sends a high-degree polynomial up on the right, a negative one sends it down. It also appears in Vieta's formulas and in the steps of polynomial division.
A small caution: the leading coefficient is the coefficient of the highest-degree term, not necessarily the first number you happen to write. If the polynomial is out of order, like -5x + 3x^2 + 7, you must first identify the highest power (x^2) — so the leading coefficient is 3, not -5.
In -2x^3 + 4x^2 - x + 9, the leading term is -2x^3, so the leading coefficient is -2.
The sign matters too — it steers the tails of the graph.
A polynomial whose leading coefficient is 1, like x^3 - 2x + 5, is called monic. Monic polynomials are convenient because their leading term needs no number out front.