Before you read

Answer first — see what you already know.

What is the degree of the polynomial $f (x) = 4 x^{3} - 2 x^{5} + x - 7$ ?

Check your answer

Answer:

Before you read

Answer first — see what you already know.

As $x \to \infty$ , what happens to $f (x) = - 3 x^{4} + x^{2} + 5$ ?

Check your answer

Answer:

Theory

Degree and Leading Coefficient

The degree of a polynomial is its highest exponent. The leading coefficient is the coefficient of that highest-degree term.

Polynomial	Degree	Leading coeff	Name
$3x + 5$	1	3	Linear
$-2x^2 + x - 1$	2	$-2$	Quadratic
$x^3 - 4x$	3	1	Cubic
$-x^4 + 2x^2 + 1$	4	$-1$	Quartic

Cubic polynomial with zeros at -2, 0, and 3

Theory

End Behavior

End behavior describes what happens to $f(x)$ as $x \to \infty$ and $x \to -\infty$ .

It depends on two things: degree (odd/even) and leading coefficient (positive/negative).

Degree	Leading coeff	Left end ( $x \to -\infty$ )	Right end ( $x \to \infty$ )
Even	Positive	$\uparrow$ (up)	$\uparrow$ (up)
Even	Negative	$\downarrow$ (down)	$\downarrow$ (down)
Odd	Positive	$\downarrow$ (down)	$\uparrow$ (up)
Odd	Negative	$\uparrow$ (up)	$\downarrow$ (down)

Memory trick: Even degree $\to$ both ends same direction. Odd degree $\to$ opposite directions. Positive leading $\to$ right end goes up.

2 more sections including examples, practice problems, and step-by-step solutions.

Try NovaMath free — AI tutoring, 115 lessons, 2,400+ exercises.