Polynomial and Rational Functions: Roots, Factorization, and Theorems

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: $f(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0$, where $a_n \neq 0$ and $n$ is a non-negative integer.

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

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

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

Example: $f(x) = x^2 - x - 2$ is a quadratic polynomial (degree 2).

Graphing Quadratic Functions

Quadratic functions have the form $f(x) = ax^2 + bx + c$. Their graphs are parabolas.

• If $a > 0$, the parabola opens upward; if $a < 0$, it opens downward.

• The vertex is at $x = -\frac{b}{2a}$.

• The axis of symmetry is the vertical line $x = -\frac{b}{2a}$.

Example: $f(x) = x^2 - x - 2$ factors as $(x-2)(x+1)$, so the roots are $x=2$ and $x=-1$.

Roots and Zeros of Polynomials

Finding Roots

The roots (or zeros) of a polynomial are the values of x for which $f(x) = 0$.

• Factoring is a common method for finding roots.

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

Example: $f(x) = x^2 - x - 2$ has roots $x = 2$ and $x = -1$.

Multiplicity of Roots

• If $(x - r)^k$ is a factor, then $r$ is a root of multiplicity $k$.

• If $k$ is even, the graph touches the x-axis at $r$; if $k$ 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 $(x - r)$.

Example: Divide $f(x) = x^3 - 6x^2 + 11x - 6$ by $x - 1$ using synthetic division.

Remainder Theorem

• If a polynomial $f(x)$ is divided by $(x - a)$, the remainder is $f(a)$.

Example: For $f(x) = x^3 - 6x^2 + 11x - 6$, $f(1) = 0$, so $x - 1$ is a factor.

Factor Theorem

The Factor Theorem states that $(x - r)$ is a factor of $f(x)$ if and only if $f(r) = 0$.

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

Example: If $f(2) = 0$, then $(x - 2)$ is a factor of $f(x)$.

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 $\pm \frac{p}{q}$, where $p$ divides the constant term and $q$ divides the leading coefficient.

Example: For $f(x) = 2x^3 + 3x^2 - 8x + 3$, possible rational roots are $\pm1, \pm3, \pm\frac{1}{2}, \pm\frac{3}{2}$.

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 $f(x) = (x-1)^2(x+2)$ touches the x-axis at $x=1$ and crosses at $x=-2$.

Summary Table: Polynomial Root Properties

Root	Multiplicity	Graph Behavior at Root
Odd	1, 3, 5, ...	Crosses x-axis
Even	2, 4, 6, ...	Touches x-axis

Applications and Examples

• Solving Polynomial Equations: Set $f(x) = 0$ and solve for $x$ 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 $f(x) = x^3 - 6x^2 + 11x - 6$, the roots are $x = 1, 2, 3$ (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.