College Algebra: Rational and Polynomial Functions Study Guide

Rational Functions

Solving Rational Equations

Rational equations involve expressions with polynomials in the numerator and denominator. To solve, find a common denominator and set the numerators equal, being careful to check for extraneous solutions.

• Key Point: Always check for values that make the denominator zero, as these are excluded from the domain.

• Example: Solve $\frac{3x^2 - 2x + 8}{2x^2 + 2}$ for $x$.

Domain of Rational Functions

The domain of a rational function is all real numbers except those that make the denominator zero.

• Key Point: Set the denominator equal to zero and solve for excluded values.

• Example: For $f(x) = \frac{2x - 12}{x^2 + 5x - 6}$, the domain excludes $x$ values that satisfy $x^2 + 5x - 6 = 0$.

Asymptotes of Rational Functions

Rational functions may have vertical, horizontal, or oblique (slant) asymptotes, which describe the behavior of the graph as $x$ approaches certain values.

• Vertical Asymptotes: Occur at values of $x$ that make the denominator zero (and not the numerator).

• Horizontal Asymptotes: Determined by comparing degrees of numerator and denominator.

• Oblique Asymptotes: Occur when the degree of the numerator is exactly one more than the denominator; found by polynomial division.

• Example: For $f(x) = \frac{2x^2 - 2x - 8}{x^2 - 2x - 15}$, vertical asymptotes are at $x$ values where $x^2 - 2x - 15 = 0$.

Polynomial Functions

Finding Polynomials with Given Zeros

To construct a polynomial with specified zeros, use the fact that each zero $r$ corresponds to a factor $(x - r)$.

• Key Point: If zeros are $a$, $b$, and $c$, the polynomial is $f(x) = k(x - a)(x - b)(x - c)$ for some constant $k$.

• Example: Find a polynomial of degree 3 with zeros 1, 2, and 4: $f(x) = (x - 1)(x - 2)(x - 4)$.

Rational Zeros Theorem

The Rational Zeros Theorem helps list all possible rational zeros of a polynomial with integer coefficients.

• Key Point: Possible rational zeros are $\pm \frac{p}{q}$, where $p$ divides the constant term and $q$ divides the leading coefficient.

• Example: For $f(x) = x^3 - 8x^2 + 17x - 36$, possible rational zeros are $\pm1, \pm2, \pm3, \pm4, \pm6, \pm12, \pm18, \pm36$.

Synthetic Division

Synthetic division is a shortcut for dividing a polynomial by a linear factor of the form $(x - c)$.

• Key Point: Use synthetic division to evaluate polynomials and find remainders quickly.

• Example: Use synthetic division to find $f(-5)$ for $f(x) = x^3 - 3x^2 + 6x - 3$.

Multiplicity of Zeros

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

• Key Point: If $(x - r)^k$ is a factor, $r$ is a zero of multiplicity $k$.

• Example: For $f(x) = (x - 3)^2(x + 2)$, $x = 3$ has multiplicity 2, $x = -2$ has multiplicity 1.

End Behavior of Polynomial Functions

The end behavior of a polynomial function is determined by its leading term.

• Key Point: For $f(x) = ax^n$, if $n$ is even and $a > 0$, both ends go up; if $n$ is odd and $a > 0$, left end down, right end up.

• Example: For $f(x) = 3x^3 - 5x^2 + 2.5$, as $x \to \infty$, $f(x) \to \infty$; as $x \to -\infty$, $f(x) \to -\infty$.

Graphing Polynomial and Rational Functions

Intercepts

Intercepts are points where the graph crosses the axes.

• x-intercept: Set $y = 0$ and solve for $x$.

• y-intercept: Set $x = 0$ and solve for $y$.

• Example: For $f(x) = 2x^2 - 3x - 2$, find $f(0)$ for the y-intercept.

Sketching Graphs

To sketch the graph of a polynomial or rational function, plot intercepts, asymptotes, and analyze end behavior and multiplicity of zeros.

• Key Point: Use zeros and their multiplicities to determine how the graph behaves at each intercept.

• Example: For $f(x) = 2x^3 + 2x^2 - 2x - 8$, plot zeros and analyze the shape.

Table: Types of Asymptotes in Rational Functions

Type	How to Find	Example
Vertical	Set denominator = 0	$x^2 - 2x - 15 = 0$
Horizontal	Compare degrees of numerator and denominator	If degrees equal, $y = \frac{a}{b}$
Oblique (Slant)	Degree numerator = degree denominator + 1; use division	$y = mx + b$

Additional info:

• Some questions involve finding critical values, which are points where the function is undefined or changes behavior.

• Multiplicity affects whether the graph crosses or touches the x-axis at a zero.

• End behavior diagrams help visualize the long-term behavior of polynomials.