Polynomial Functions and Their Graphs

Definition and Key Concepts

Polynomial functions are a central topic in precalculus, characterized by expressions involving powers of x with real coefficients. Understanding their structure, zeros, and graphical behavior is essential for further study in mathematics.

• Polynomial of Degree n: A function of the form , where and is a non-negative integer.

• Leading Coefficient: The coefficient of the highest power of in the polynomial.

• Examples: ,

• Non-Examples: Expressions that are not polynomials, such as those with negative or fractional exponents, or variables in denominators.

End Behavior of Polynomial Functions

The end behavior of a polynomial function describes how the function behaves as approaches positive or negative infinity. This is determined by the degree and the leading coefficient of the polynomial.

• Lead Coefficient Test:

  • If is even and , both ends of the graph rise ( as ).

  • If is even and , both ends of the graph fall ( as ).

  • If is odd and , left end falls and right end rises ( as , as ).

  • If is odd and , left end rises and right end falls ( as , as ).

• Example: For , (even), (negative), so both ends fall.

Finding Zeros of Polynomial Functions

Factoring and Solving

Zeros (or roots) of a polynomial are the values of for which . Factoring is a primary method for finding these zeros.

• Example:

• Example:

Multiplicity of Zeros

The multiplicity of a zero refers to the number of times a particular root appears in the factorization of the polynomial.

• Even Multiplicity: The graph touches the x-axis at the zero but does not cross it.

• Odd Multiplicity: The graph crosses the x-axis at the zero.

• Example: For , if is a root of multiplicity 2, the graph touches the axis at $x=0$.

Advanced Theorems and Techniques

Fundamental Theorem of Algebra

The Fundamental Theorem of Algebra states that every non-constant polynomial function of degree has exactly $n$ complex roots (counting multiplicities).

Linear Factorization Theorem

Every polynomial of degree can be factored into $n$ linear factors over the complex numbers:

• , where are the (possibly complex) roots.

Rational Root Test

The Rational Root Test provides a way to find all possible rational roots of a polynomial equation with integer coefficients. Possible rational roots are of the form , where divides the constant term and divides the leading coefficient.

• Example: For , possible rational roots are .

Polynomial Division

Long Division of Polynomials

When a polynomial cannot be factored easily, long division is used to divide polynomials and find factors or simplify expressions.

• Example: Divide by a linear or quadratic factor to find zeros.

• Example:

Summary Table: Key Properties of Polynomial Functions