Polynomial and Rational Functions: Definitions, Properties, and Theorems

Polynomial Functions

Definition and General Form

A polynomial function in one variable is a function of the form:

$f(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0$

• Degree: The highest power of $x$ with a nonzero coefficient ($n$).

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

• Domain: All real numbers, $(-\infty, \infty)$.

Example: $f(x) = 4x^3 - 5x^2 + 2x - 7$ is a polynomial of degree 3 with leading coefficient 4.

Summary of Properties

Degree	Form	Name	Graph
0	$f(x) = a_0$	Zero function	Horizontal line
1	$f(x) = a_1x + a_0$	Linear function	Straight line
2	$f(x) = a_2x^2 + a_1x + a_0$	Quadratic function	Parabola
3	$f(x) = a_3x^3 + \cdots$	Cubic function	S-shaped curve
n	$f(x) = a_nx^n + \cdots$	nth-degree polynomial	Varies, smooth and continuous

Polynomial graphs are always smooth and continuous (no breaks, holes, or sharp corners).

Zeros and Multiplicity

A zero (or root) of a polynomial $f(x)$ is a value $r$ such that $f(r) = 0$.

• If $(x - r)^m$ is a factor of $f(x)$ and $(x - r)^{m+1}$ is not, then $r$ is a zero of multiplicity $m$.

• If $m$ is odd, the graph crosses the x-axis at $r$; if $m$ is even, the graph touches but does not cross.

Example: $f(x) = (x - 3)^2(x + 4)$ has a zero at $x = 3$ (multiplicity 2) and $x = -4$ (multiplicity 1).

Turning Points

• A polynomial of degree $n$ has at most $n - 1$ turning points (local maxima or minima).

• Turning points are where the graph changes direction.

End Behavior

The end behavior of a polynomial function is determined by its leading term $a_nx^n$:

• If $n$ is even and $a_n > 0$, both ends rise; if $a_n < 0$, both ends fall.

• If $n$ is odd and $a_n > 0$, left end falls and right end rises; if $a_n < 0$, left end rises and right end falls.

The Real Zeros of a Polynomial Function

Division Algorithm and Remainder Theorem

• Division Algorithm: For polynomials $f(x)$ and $d(x) \neq 0$, there exist unique polynomials $q(x)$ and $r(x)$ such that $f(x) = d(x)q(x) + r(x)$, where $r(x) = 0$ or $ext{deg}(r) < ext{deg}(d)$.

• Remainder Theorem: The remainder when $f(x)$ is divided by $x - c$ is $f(c)$.

Factor Theorem

• $x - c$ is a factor of $f(x)$ if and only if $f(c) = 0$.

Number of Real Zeros and Descartes' Rule of Signs

• A polynomial of degree $n$ has at most $n$ real zeros.

• Descartes' Rule of Signs: The number of positive real zeros equals the number of sign changes in $f(x)$ or less by an even integer. The number of negative real zeros equals the number of sign changes in $f(-x)$ or less by an even integer.

Rational Zeros Theorem

For $f(x) = a_nx^n + \cdots + a_0$ with integer coefficients, any rational zero $\frac{p}{q}$ (in lowest terms) must have $p$ dividing $a_0$ and $q$ dividing $a_n$.

Intermediate Value Theorem

If $f(a)$ and $f(b)$ have opposite signs, then $f(x)$ has at least one real zero between $a$ and $b$.

Complex Zeros and the Fundamental Theorem of Algebra

Complex Zeros

• A complex zero is a solution to $f(x) = 0$ where $x$ is a complex number.

• Every polynomial of degree $n \geq 1$ has exactly $n$ complex zeros (counting multiplicity).

• If coefficients are real, non-real zeros occur in conjugate pairs.

Factoring Polynomials

• Every polynomial with real coefficients can be factored into linear and/or irreducible quadratic factors over the real numbers.

Rational Functions

Definition and Domain

A rational function is a function of the form $R(x) = \frac{p(x)}{q(x)}$, where $p$ and $q$ are polynomials and $q(x) \neq 0$.

• The domain is all real numbers except where $q(x) = 0$.

Asymptotes

• Vertical Asymptote: $x = a$ is a vertical asymptote if $q(a) = 0$ and $p(a) \neq 0$.

• Horizontal Asymptote: $y = L$ is a horizontal asymptote if $\lim_{x \to \pm\infty} R(x) = L$.

• Oblique (Slant) Asymptote: Occurs if the degree of the numerator is exactly one more than the denominator.

Finding Asymptotes

• If degree of numerator < degree of denominator: horizontal asymptote at $y = 0$.

• If degrees are equal: horizontal asymptote at $y = \frac{\text{leading coefficient of numerator}}{\text{leading coefficient of denominator}}$.

• If degree of numerator > degree of denominator by 1: oblique asymptote found by polynomial division.

Polynomial and Rational Inequalities

Solving Inequalities

• Set the inequality to zero and factor the numerator and denominator.

• Find the zeros and undefined points to divide the real line into intervals.

• Test each interval to determine where the inequality holds.

Example: Solve $\frac{2x^2 + 5x + 6}{(x + 1)^2} \geq 2$ by finding critical points and testing intervals.

Summary Table: Key Theorems and Properties

Theorem/Property	Description
Division Algorithm	Expresses $f(x)$ as $d(x)q(x) + r(x)$
Remainder Theorem	Remainder of $f(x)$ divided by $x-c$ is $f(c)$
Factor Theorem	$x-c$ is a factor of $f(x)$ iff $f(c)=0$
Descartes' Rule of Signs	Predicts number of positive/negative real zeros
Rational Zeros Theorem	Possible rational zeros are $\pm\frac{p}{q}$
Intermediate Value Theorem	If $f(a)$ and $f(b)$ have opposite signs, a zero exists between $a$ and $b$
Fundamental Theorem of Algebra	Degree $n$ polynomial has $n$ complex zeros
Conjugate Pairs Theorem	Non-real zeros occur in conjugate pairs

Additional info: These notes are based on textbook-style explanations and include both definitions and worked examples, as well as summary tables and graphical interpretations, suitable for Precalculus students studying polynomial and rational functions.