What is a leading coefficient?

The leading coefficient is the coefficient of the term with the highest degree in a polynomial. In a polynomial function written in standard form, the leading coefficient is the number that appears in front of the term with the highest exponent.

The leading coefficient plays a key role in determining the behavior of a polynomial function. It helps determine the direction in which the graph of the polynomial opens, and it also affects the end behavior of the function as x approaches positive or negative infinity.

For example, in the polynomial function f(x) = 3x^4 - 2x^2 + 5x + 1, the leading coefficient is 3, as it is the coefficient of the term with the highest degree (x^4). This means that the graph of the function will open upwards and that as x approaches positive or negative infinity, the function will also approach positive or negative infinity, respectively.

Overall, the leading coefficient provides important information about the overall shape and behavior of a polynomial function.