### Notes on Polynomial Functions

#### 1. Definition and General Form

- A polynomial function is a function that can be expressed in the form:

\[

P(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0

\]

where:

- \( n \) is a non-negative integer (the degree of the polynomial),

- \( a_n, a_{n-1}, \ldots, a_0 \) are constants (coefficients),

- \( a_n \neq 0 \).

#### 2. Types of Polynomial Functions

- Constant Polynomial: \( P(x) = a_0 \) (degree 0)

- Linear Polynomial: \( P(x) = a_1x + a_0 \) (degree 1)

- Quadratic Polynomial: \( P(x) = a_2x^2 + a_1x + a_0 \) (degree 2)

- Cubic Polynomial: \( P(x) = a_3x^3 + a_2x^2 + a_1x + a_0 \) (degree 3)

- Quartic Polynomial: \( P(x) = a_4x^4 + a_3x^3 + a_2x^2 + a_1x + a_0 \) (degree 4)

#### 3. Properties

- Degree: The highest power of \( x \) with a non-zero coefficient.

- Leading Coefficient: The coefficient of the term with the highest degree.

- Constant Term: The term \( a_0 \).

#### 4. Operations with Polynomials

- Addition/Subtraction: Combine like terms.

- Multiplication: Distribute each term in the first polynomial by each term in the second polynomial.

- Division: Polynomial long division or synthetic division (for divisors of the form \( x - c \)).

#### 5. Roots/Zeros of Polynomial Functions

- Root: A solution to the equation \( P(x) = 0 \).

- Factor Theorem: If \( c \) is a root of \( P(x) \), then \( x - c \) is a factor of \( P(x) \).

- Multiplicity: If a root \( r \) is repeated \( k \) times in the factorization, it has multiplicity \( k \).

#### 6. Graphing Polynomial Functions

- End Behavior: Determined by the leading term \( a_nx^n \):

- If \( n \) is even and \( a_n > 0 \), \( P(x) \) rises to both left and right.

- If \( n \) is even and \( a_n < 0 \), \( P(x) \) falls to both left and right.

- If \( n \) is odd and \( a_n > 0 \), \( P(x) \) falls to the left and rises to the right.

- If \( n \) is odd and \( a_n < 0 \), \( P(x) \) rises to the left and falls to the right.

- Turning Points: A polynomial of degree \( n \) can have up to \( n-1 \) turning points.

- Intercepts: The points where the polynomial crosses the x-axis (roots) and y-axis (constant term).

### Problems to Study

1. Find the Degree and Leading Coefficient

- Determine the degree and leading coefficient of \( P(x) = 4x^5 - 3x^3 + 2x^2 - x + 7 \).

2. Addition and Subtraction

- Add and subtract \( P(x) = 3x^4 + 2x^3 - x + 1 \) and \( Q(x) = -2x^4 + x^3 + 5x - 3 \).

3. Multiplication

- Multiply \( P(x) = 2x^2 + 3x + 1 \) and \( Q(x) = x - 4 \).

4. Division

- Divide \( P(x) = 4x^3 - 6x^2 + 2x - 1 \) by \( Q(x) = x - 1 \) using polynomial long division.

5. Finding Roots

- Find all the roots of \( P(x) = x^3 - 6x^2 + 11x - 6 \).

- Determine the multiplicity of each root.

6. Factorization

- Factor \( P(x) = x^4 - 5x^3 + 6x^2 + 4x - 8 \) completely.

7. Graphing

- Sketch the graph of \( P(x) = -x^3 + 3x^2 - 4x + 2 \). Identify the end behavior, turning points, and intercepts.

8. Applications

- A rectangular garden has a length that is 3 meters longer than its width. If the area of the garden is \( 4 \text{ m}^2 \), find the dimensions of the garden by forming and solving a polynomial equation.

By studying these notes and solving the problems, you'll gain a solid understanding of polynomial functions and their properties.