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Polynomial Functions — Degree and End Behavior

22 min Quadratic and Polynomial Functions

Polynomial Functions

Polynomial functions extend quadratics to higher degrees — cubics, quartics, and beyond. The behavior of a polynomial is governed by its degree (the highest power) and its leading coefficient, which together determine the end behavior of the graph.

A polynomial of degree n can have at most n real zeros (x-intercepts) and at most n − 1 turning points. These limits let you sketch a reasonable graph even before plotting individual points.

This lesson introduces polynomial terminology, degree, leading coefficient, and end behavior, and teaches you to identify the basic shape of a polynomial graph from its equation.

Definition

A polynomial of degree $n$: $P(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0$ where $a_n \neq 0$.

$a_n$ is the leading coefficient; $n$ is the degree.

End Behavior

As $x \to \pm\infty$, the leading term $a_n x^n$ dominates.

Degree	$a_n$	Left	Right
Even	$> 0$	$\uparrow$	$\uparrow$
Even	$< 0$	$\downarrow$	$\downarrow$
Odd	$> 0$	$\downarrow$	$\uparrow$
Odd	$< 0$	$\uparrow$	$\downarrow$

Zeros and Multiplicity

Behavior at Zeros

If $(x - r)^k$ is a factor:

$k$ odd: graph crosses the $x$-axis at $r$
$k$ even: graph touches (bounces) at $r$

Example 1

$P(x) = (x - 1)^2(x + 3)$. Find degree, end behavior, and zeros.

Degree $= 3$ (odd), leading coeff $= 1 > 0$. End: $\downarrow$ left, $\uparrow$ right.

Zeros: $x = 1$ (multiplicity 2, bounces), $x = -3$ (mult. 1, crosses).

Example 2

$P(x) = -2x^4 + 5x^3 - x$. End behavior?

Degree $4$ (even), $a_n = -2 < 0$. Both ends go $\downarrow$.

Example 3

$P(x) = x(x+2)^2(x-3)$. Zeros and behavior at each.

$x = 0$ (crosses), $x = -2$ (bounces), $x = 3$ (crosses). Degree $= 4$.

Practice Problems

1. Degree of $4x^5 - 3x^2 + 1$

2. End behavior of $-x^3 + 2x$

3. End behavior of $x^4 - x^2$

4. Zeros and mult. of $(x-2)^3(x+1)^2$

5. Cross or bounce at each zero of $x^2(x-4)$

6. Degree and leading coeff of $-3(x+1)^2(x-5)$

7. Max turning points for degree 4 polynomial

8. End behavior of $2x^6 - x^3 + 7$

9. How many $x$-intercepts can a degree 3 polynomial have at most?

10. Find zeros: $x^3 - 4x^2 + 4x = 0$

Show Answer Key

1. $5$

2. Odd, $a_n < 0$: $\uparrow$ left, $\downarrow$ right

3. Even, $a_n > 0$: $\uparrow$ both ends

4. $x = 2$ (mult 3, crosses), $x = -1$ (mult 2, bounces)

5. $x = 0$ bounces (mult 2), $x = 4$ crosses (mult 1)

6. Degree $3$, leading coeff $= -3$

7. At most $3$ turning points

8. Even, $a_n > 0$: $\uparrow$ both ends

9. At most $3$

10. $x(x^2-4x+4) = x(x-2)^2 = 0$; $x = 0$ or $x = 2$

Graphing Quadratics — Vertex Form Remainder and Factor Theorems

Polynomial Functions — Degree and End Behavior