Polynomial Functions and Their Properties

Definition and Basic Properties

Polynomial functions are algebraic expressions involving sums of powers of x with constant coefficients. The general form is:

• Definition: , where and is a non-negative integer.

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

• Leading Coefficient: The coefficient of the highest degree term.

• End Behavior: Determined by the degree and leading coefficient.

Example: is a quadratic polynomial (degree 2).

Graphing Quadratic Functions

Quadratic functions have the form . Their graphs are parabolas.

• If , the parabola opens upward; if , it opens downward.

• The vertex is at .

• The axis of symmetry is the vertical line .

Example: factors as , so the roots are and .

Roots and Zeros of Polynomials

Finding Roots

The roots (or zeros) of a polynomial are the values of x for which .

• Factoring is a common method for finding roots.

• For higher-degree polynomials, use the Rational Root Theorem or synthetic division.

Example: has roots and .

Multiplicity of Roots

• If is a factor, then is a root of multiplicity .

• If is even, the graph touches the x-axis at ; if is odd, it crosses the x-axis.

Polynomial Division and the Remainder Theorem

Long Division and Synthetic Division

Polynomials can be divided using long division or synthetic division.

• Long Division: Similar to numerical long division, used for dividing any polynomials.

• Synthetic Division: A shortcut for dividing by linear factors of the form .

Example: Divide by using synthetic division.

Remainder Theorem

• If a polynomial is divided by , the remainder is .

Example: For , , so is a factor.

Factor Theorem

The Factor Theorem states that is a factor of if and only if .

• Use this theorem to test possible rational roots and factor polynomials completely.

Example: If , then is a factor of .

Rational Root Theorem

The Rational Root Theorem provides a way to list all possible rational roots of a polynomial 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 .

Graphing Higher-Degree Polynomials

To graph a polynomial, consider its degree, leading coefficient, roots, and their multiplicities.

• End behavior: For even degree, both ends go in the same direction; for odd degree, ends go in opposite directions.

• Multiplicity affects how the graph interacts with the x-axis at each root.

Example: The graph of touches the x-axis at and crosses at .

Summary Table: Polynomial Root Properties

Applications and Examples

• Solving Polynomial Equations: Set and solve for using factoring, synthetic division, or the Rational Root Theorem.

• Graphing: Identify roots, their multiplicities, and end behavior to sketch the graph.

• Checking Factors: Use the Factor Theorem and synthetic division to verify if a binomial is a factor.

Example: For , the roots are (all multiplicity 1), so the graph crosses the x-axis at each root.

Additional info: These notes also include worked examples of synthetic division, tables for possible rational roots, and step-by-step solutions for factoring and finding all real roots of cubic and quartic polynomials.