Leading coefficient

Definition

The leading coefficient of a polynomial is the coefficient of the term with the highest degree. It plays a crucial role in determining the end behavior of the polynomial function.

5 Must Know Facts For Your Next Test

1. The leading coefficient can be found by identifying the term with the highest exponent in a polynomial and noting its coefficient.
2. It affects the end behavior of the graph of a polynomial function; if positive, both ends go up for even degree, and one end goes up and one goes down for odd degree polynomials.
3. In standard form, polynomials are written with terms in descending order of their exponents; thus, the leading coefficient appears first.
4. For even-degree polynomials, if the leading coefficient is negative, both ends of the graph will go down.
5. For odd-degree polynomials, if the leading coefficient is negative, one end of the graph will go down while the other goes up.

Review Questions

• How does changing the sign of the leading coefficient affect the end behavior of an even-degree polynomial?
• What is the leading coefficient in the polynomial $3x^4 - 5x^3 + x - 7$?
• Explain how to find and interpret the leading coefficient in any given polynomial.

Related terms

Polynomial:

An expression consisting of variables and coefficients, involving only addition, subtraction, multiplication, and non-negative integer exponents.

Degree:

The highest exponent in a polynomial expression; it determines many properties of its graph.

$y$-intercept: The point where a graph intersects with the $y$-axis; found by evaluating a polynomial at $x = 0$.