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 is found by identifying the term with the highest power in a polynomial and noting its coefficient.
2. If the leading coefficient is positive, the graph of the polynomial will generally rise to infinity as $x$ approaches positive or negative infinity for even degrees.
3. For odd degree polynomials, a positive leading coefficient means the graph will fall to negative infinity as $x$ approaches negative infinity and rise to positive infinity as $x$ approaches positive infinity.
4. The magnitude of the leading coefficient affects how steep or flat the graph of the polynomial appears.
5. In standard form, a polynomial is written with terms in descending order by degree, making it easier to identify the leading coefficient.

Review Questions

• How do you determine which term's coefficient is the leading coefficient in a polynomial?
• What effect does a negative leading coefficient have on an odd-degree polynomial's end behavior?
• Why is it important to write a polynomial in standard form when identifying its leading coefficient?

Related terms

Degree of a Polynomial: The highest power/exponent of any term in the polynomial.

Standard Form: A way of writing down very large or very small numbers easily; for polynomials, it’s writing terms in descending order by degree.

End Behavior: The direction that the graph heads as $x$ approaches positive or negative infinity.