Polynomial Functions

Definition and Structure of Polynomials

Polynomial functions are algebraic expressions consisting of variables and coefficients, involving only non-negative integer powers of the variable. The general form is:

• Definition: A polynomial of degree n is given by , where .

• Degree: The highest power of x in the polynomial.

• Leading Term: The term with the highest degree.

• Leading Coefficient: The coefficient of the leading term.

• Constant Term: The term without a variable ().

Example: For , the leading term is , the leading coefficient is 6, and the degree is 4.

Classification of Polynomials

• Constant: Degree 0 (e.g., )

• Linear: Degree 1 (e.g., )

• Quadratic: Degree 2 (e.g., )

• Cubic: Degree 3 (e.g., )

• Quartic: Degree 4 (e.g., )

End Behavior of Polynomial Functions

Understanding End Behavior

The end behavior of a polynomial function describes how the function behaves as or . It is determined by the leading term.

• Even Degree, Positive Leading Coefficient: Both ends rise ().

• Even Degree, Negative Leading Coefficient: Both ends fall ().

• Odd Degree, Positive Leading Coefficient: Left falls, right rises.

• Odd Degree, Negative Leading Coefficient: Left rises, right falls.

Example: For , the end behavior is both ends up.

Graphing Polynomial Functions

Intercepts and Turning Points

• x-intercepts: Points where .

• y-intercept: Point where .

• Turning Points: Points where the graph changes direction. A polynomial of degree has at most turning points.

Example: For , the x-intercepts are found by solving .

Multiplicity of Zeros

• Multiplicity: The number of times a zero occurs. If is a factor, is a zero of multiplicity .

• Behavior: If multiplicity is odd, the graph crosses the x-axis at . If even, it touches and turns around.

Example: For , is a zero with multiplicity 2, $1-6$ with multiplicity 1.

Analyzing Polynomial Functions

Finding Zeros and Using Substitution

• To check if is a zero, substitute into and see if .

• Synthetic Division: A shortcut for dividing polynomials by to find the quotient and remainder.

Example: Use synthetic division to divide by .

Maximum Number of Real Zeros and Turning Points

• A polynomial of degree has at most real zeros and turning points.

• Not all zeros or turning points may be real; some may be complex.

Rational Functions

Definition and Structure

A rational function is a ratio of two polynomials:

• Definition: , where .

Domain and Asymptotes

• Domain: All real numbers except where .

• Vertical Asymptotes: Values of where and .

• Horizontal Asymptotes: Determined by the degrees of and :

  • If degree of < degree of , is the horizontal asymptote.

  • If degrees are equal, .

  • If degree of > degree of , no horizontal asymptote (may have an oblique asymptote).

Example: For , vertical asymptote at , horizontal asymptote at .

Intercepts and Graphing

• x-intercept: Set numerator equal to zero, solve for .

• y-intercept: Set and evaluate .

• Plot at least three points for an accurate graph.

Inverse Functions

Finding the Inverse

• To find the inverse , solve for in terms of , then swap and .

Example: For , solve for :

• So,

Tables: Properties and Analysis

Sample Table: Maximums and Turning Points

Sample Table: Intervals of Positivity/Negativity

Summary of Key Concepts

• Polynomials are classified by degree and leading coefficient.

• End behavior is determined by the leading term.

• Zeros and their multiplicities affect the graph's shape.

• Rational functions have domains restricted by the denominator and may have vertical/horizontal asymptotes.

• Inverse functions are found by solving for and swapping variables.

• Tables help analyze intervals of positivity/negativity and function properties.

Additional info: Some content inferred from context and standard Precalculus curriculum, including definitions, examples, and table structure.