D5

Polynomial Functions

Introduction to Polynomial Functions

Polynomial functions are a central topic in College Algebra, characterized by equations of the form $f(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0$, where $n$ is a non-negative integer and coefficients $a_i$ are real numbers. Understanding their graphs, roots, and end behavior is essential for analyzing and solving algebraic problems.

• Degree: The highest power of $x$ in the polynomial. Determines the general shape and complexity of the graph.

• Leading Coefficient: The coefficient of the term with the highest degree. Influences the end behavior of the graph.

• Roots/Zeros: Values of $x$ where $f(x) = 0$. These are the x-intercepts of the graph.

Characteristics of Polynomial Graphs

• Turning Points: A polynomial of degree $n$ can have at most $n-1$ turning points (local maxima or minima).

• Intercepts: The graph can have up to $n$ x-intercepts, but may have fewer if some roots are repeated or complex.

• End Behavior: Determined by the degree and leading coefficient. For large $|x|$, the graph resembles the leading term $a_nx^n$.

End Behavior of Polynomial Functions

The end behavior describes how the function behaves as $x \to \infty$ or $x \to -\infty$. It depends on both the degree and the sign of the leading coefficient.

Degree (n)	Leading Coefficient	End Behavior	Example
Even	Positive	Both ends up	$y = x^2$
Even	Negative	Both ends down	$y = -x^2$
Odd	Positive	Left down, right up	$y = x^3$
Odd	Negative	Left up, right down	$y = -x^3$

Roots and Multiplicity

• Root (Zero): If $f(r) = 0$, then $r$ is a root of $f(x)$.

• Multiplicity: The number of times a root is repeated. If $(x - r)^k$ is a factor, $r$ is a root of multiplicity $k$.

• Graph Behavior at Roots:

  • If multiplicity is odd, the graph crosses the x-axis at the root.

  • If multiplicity is even, the graph touches but does not cross the x-axis at the root.

Factor Theorem

The Factor Theorem states that $x - a$ is a factor of $f(x)$ if and only if $f(a) = 0$. This is useful for finding roots and factoring polynomials.

Examples and Applications

• Example 1: For $f(x) = x^3 - 2x^2 - x + 2$, the degree is 3, leading coefficient is 1, and the graph has up to 3 roots and 2 turning points.

• Example 2: For $f(x) = (x-2)^2(x+1)$, degree is 3, root at $x=2$ with multiplicity 2 (touches x-axis), root at $x=-1$ with multiplicity 1 (crosses x-axis).

• Example 3: For $f(x) = -2x^3$, degree is 3 (odd), leading coefficient is negative, so left end goes up, right end goes down.

Maximum Number of Roots and Turning Points

Degree	Maximum Number of Roots	Maximum Number of Turning Points
1	1	0
2	2	1
3	3	2
4	4	3

Intercepts

• x-intercepts: Set $f(x) = 0$ and solve for $x$.

• y-intercept: Set $x = 0$ and compute $f(0)$.

• Example: For $f(x) = x^2 - 5x + 6$, x-intercepts are $x=2$ and $x=3$; y-intercept is $f(0) = 6$.

Sketching Polynomial Graphs

To sketch a polynomial graph, follow these steps:

1. Identify the degree and leading coefficient to determine end behavior.

2. Find all real roots and their multiplicities.

3. Calculate the y-intercept.

4. Plot turning points (if possible) and sketch the general shape.

Practice Problems (Summary)

• Determine end behavior using the Leading Coefficient Test.

• Complete tables for degree, maximum roots, and turning points.

• Find roots, intercepts, and sketch graphs for given polynomials.

Additional info:

• Some notes include handwritten clarifications and examples, reinforcing the concepts of degree, roots, multiplicity, and end behavior.

• Tables and graphs are used to compare polynomial functions of different degrees and coefficients.