Leading coefficient

Definition

The leading coefficient is the coefficient of the term with the highest degree in a polynomial. It determines the polynomial's end behavior and influences its graph's shape.

5 Must Know Facts For Your Next Test

1. In a polynomial $f(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0$, the leading coefficient is $a_n$.
2. The sign of the leading coefficient affects whether the ends of the graph rise or fall.
3. For higher-degree polynomials, the leading coefficient primarily dictates the graph's end behavior as $|x|$ becomes large.
4. If the leading coefficient is positive, for even-degree polynomials, both ends of the graph rise; for odd-degree polynomials, one end rises and one falls.
5. If the leading coefficient is negative, for even-degree polynomials, both ends of the graph fall; for odd-degree polynomials, one end falls and one rises.

Review Questions

• What role does the leading coefficient play in determining a polynomial's end behavior?
• How does changing the sign of the leading coefficient affect an even-degree polynomial's graph?
• $f(x) = -3x^4 + x^3 - x + 2$: Identify its leading coefficient.

Related terms

Polynomial: A mathematical expression consisting of variables (or indeterminates), coefficients, and non-negative integer exponents of variables.

Degree:

The highest power of the variable in a polynomial.

End Behavior:

The direction that the graph of a function heads as $x$ approaches positive or negative infinity.