Polynomial and Rational Functions: Key Concepts and Techniques

Chapter 4: Polynomial Functions and Rational Functions

4.1 Polynomial Functions and Modeling

Polynomial functions are a fundamental class of functions in precalculus, characterized by their algebraic structure and predictable end behavior. Understanding their properties is essential for graphing and solving equations.

• Definition: A polynomial function of degree n is given by: $P(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0$ where the coefficients $a_n, a_{n-1}, \ldots, a_0$ are real numbers and the exponents are whole numbers.

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

• End Behavior: The leading term $a_nx^n$ determines the end behavior of the graph as $x \to \pm\infty$.

• End Behavior Cases:

  1. Positive leading coefficient with even degree: both ends up.

  2. Negative leading coefficient with even degree: both ends down.

  3. Positive leading coefficient with odd degree: left end down, right end up.

  4. Negative leading coefficient with odd degree: left end up, right end down.

4.2 Zeros and Multiplicity of Polynomial Functions

The zeros of a polynomial function are the values of $x$ for which $P(x) = 0$. These are also called roots or solutions.

• Finding Zeros: Set $P(x) = 0$ and solve for $x$.

• Multiplicity: If a factor $(x - r)$ appears $k$ times in the factorization, $r$ is a zero of multiplicity $k$.

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

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

• Example: $f(x) = (x-2)^2(x+1)$ has a zero at $x=2$ (multiplicity 2, even) and $x=-1$ (multiplicity 1, odd).

4.3 Graphing Polynomial Functions

Graphing polynomials involves analyzing their degree, zeros, multiplicities, and end behavior.

• Number of Zeros: A degree $n$ polynomial has exactly $n$ zeros (real or complex, counting multiplicity).

• Turning Points: The graph has at most $n-1$ turning points (local maxima or minima).

• Steps to Sketch:

  1. Determine the degree and leading coefficient.

  2. Find zeros and their multiplicities.

  3. Determine end behavior.

  4. Plot intercepts and turning points.

  5. Sketch the curve, using the above information.

4.4 Theorems about Zeros of Polynomial Functions

Several theorems help in finding and understanding the zeros of polynomials.

• Intermediate Value Theorem: If $P(a)$ and $P(b)$ have opposite signs, there is at least one real zero between $a$ and $b$.

• Factor Theorem: If $f(c) = 0$, then $(x-c)$ is a factor of $f(x)$.

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

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

• Conjugate Pairs: Irrational and non-real zeros occur in conjugate pairs if the polynomial has real coefficients.

• Rational Zero Theorem: If $\frac{p}{q}$ is a rational zero of $f(x)$ with integer coefficients, $p$ divides the constant term and $q$ divides the leading coefficient.

4.5 Polynomial Division: Long and Synthetic Division

Polynomial division is used to simplify expressions and find factors or remainders.

• Long Division: Used to divide any polynomial by another of lower degree.

• Synthetic Division: A shortcut for dividing by linear factors of the form $(x-c)$.

• Example: Divide $x^3 + 7x^2 + 2x - 40$ by $x + 5$ using long division.

4.6 Rational Functions

A rational function is a quotient of two polynomials. Their domains, asymptotes, and holes are determined by the numerator and denominator.

• Definition: $f(x) = \frac{p(x)}{q(x)}$, where $q(x) \neq 0$.

• Domain: All real $x$ such that $q(x) \neq 0$.

• Vertical Asymptotes: Occur at zeros of $q(x)$ (unless canceled by a factor in $p(x)$).

• Horizontal Asymptotes: Determined by the degrees of $p(x)$ and $q(x)$:

  • If degree numerator < degree denominator: $y = 0$

  • If degree numerator = degree denominator: $y = \frac{a}{b}$ (ratio of leading coefficients)

  • If degree numerator > degree denominator: No horizontal asymptote

• Oblique (Slant) Asymptotes: Occur if degree numerator is exactly one more than degree denominator.

• Holes: Occur where a factor cancels in both numerator and denominator.

4.7 Polynomial and Rational Inequalities

Solving inequalities involving polynomials and rational functions requires finding zeros and analyzing sign changes.

• Steps for Polynomial Inequalities:

  1. Write the inequality in standard form.

  2. Solve $P(x) = 0$ to find boundary points.

  3. Test intervals between boundary points.

  4. Use a sign chart or graph to determine solution intervals.

• Steps for Rational Inequalities:

  1. Write the inequality in the form $\frac{p(x)}{q(x)} \gtrless 0$.

  2. Solve $p(x) = 0$ and $q(x) = 0$ for boundary points.

  3. Test intervals and exclude points where $q(x) = 0$ (undefined).

Summary Table: Asymptotes of Rational Functions

Type	How to Find	Effect on Graph
Vertical Asymptote	Set denominator $= 0$ (after canceling common factors)	Graph approaches infinity near this $x$-value
Horizontal Asymptote	Compare degrees of numerator and denominator	Graph levels off as $x \to \pm\infty$
Oblique Asymptote	Degree numerator = degree denominator + 1	Graph approaches a slant line as $x \to \pm\infty$
Hole	Common factor in numerator and denominator	Point missing from the graph

Example Problems

• Find the zeros and their multiplicities for $f(x) = (x-2)^2(x+1)$. Zeros: $x=2$ (multiplicity 2), $x=-1$ (multiplicity 1).

• Determine the vertical and horizontal asymptotes for $f(x) = \frac{x^2-1}{x^2-4}$. Vertical asymptotes: $x=2$, $x=-2$; Horizontal asymptote: $y=1$.

Additional info: These notes are structured to provide a comprehensive overview of polynomial and rational functions, including their properties, graphing techniques, and key theorems relevant to a Precalculus course.