Precalculus Study Notes: Complex Numbers, Quadratic, Polynomial, and Rational Functions

2.1 Complex Numbers

Complex Numbers and Imaginary Numbers

Complex numbers extend the real number system by including the imaginary unit. This allows for the solution of equations that have no real solutions.

• Imaginary Unit: The imaginary unit is defined as $i = \sqrt{-1}$, so $i^2 = -1$.

• Complex Numbers: Any number of the form $a + bi$, where $a$ and $b$ are real numbers, is a complex number.

• Standard Form: The standard form of a complex number is $a + bi$.

Operations on Complex Numbers

Complex numbers can be added, subtracted, multiplied, and divided using algebraic rules, similar to binomials.

• Addition/Subtraction: Combine like terms: $(a + bi) + (c + di) = (a + c) + (b + d)i$.

• Multiplication: Use distributive property and $i^2 = -1$.

• Conjugate: The conjugate of $a + bi$ is $a - bi$. The product of a complex number and its conjugate is always a real number: $(a + bi)(a - bi) = a^2 + b^2$.

• Division: To divide by a complex number, multiply numerator and denominator by the conjugate of the denominator.

Principal Square Root of a Negative Number

• For any positive real number $b$, $\sqrt{-b} = i\sqrt{b}$.

Quadratic Equations with Complex Imaginary Solutions

Quadratic equations with negative discriminants have complex solutions.

• General quadratic: $ax^2 + bx + c = 0$

• Quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

• If $b^2 - 4ac < 0$, solutions are complex conjugates.

2.2 Quadratic Functions

The Standard Form of a Quadratic Function

A quadratic function is a polynomial of degree 2 and can be written as:

• Standard form: $f(x) = a(x - h)^2 + k$, $a \neq 0$

• Vertex: $(h, k)$

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

Graphing Quadratic Functions

1. Determine if the parabola opens upward ($a > 0$) or downward ($a < 0$).

2. Find the vertex $(h, k)$.

3. Find the $x$-intercepts by solving $f(x) = 0$.

4. Find the $y$-intercept by evaluating $f(0)$.

5. Find the axis of symmetry: $x = h$.

6. Find the domain (all real numbers) and range (depends on $k$ and $a$).

Minimum and Maximum Values

• If $a > 0$, the minimum value occurs at $x = -\frac{b}{2a}$.

• If $a < 0$, the maximum value occurs at $x = -\frac{b}{2a}$.

• The minimum or maximum value is $f\left(-\frac{b}{2a}\right)$.

2.3 Polynomial Functions and Their Graphs

Definition of a Polynomial Function

A polynomial function is an expression of the form $f(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0$, where $n$ is a nonnegative integer and $a_n \neq 0$.

• Degree: The highest power of $x$.

• Leading Coefficient: The coefficient of the highest power of $x$.

Graphs of Polynomial Functions

• Polynomial functions of degree 2 or higher have graphs that are smooth (no sharp corners) and continuous (no breaks).

The Leading Coefficient Test

The Leading Coefficient Test determines the end behavior of a polynomial function:

• If $n$ is even and $a_n > 0$, both ends rise.

• If $n$ is even and $a_n < 0$, both ends fall.

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

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

Zeros of Polynomial Functions

• Zeros: Values of $x$ for which $f(x) = 0$.

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

• If $k$ is even, the graph touches the $x$-axis and turns around at $r$.

• If $k$ is odd, the graph crosses the $x$-axis at $r$.

A Strategy for Graphing Polynomial Functions

1. Use the Leading Coefficient Test for end behavior.

2. Find $x$-intercepts (zeros) and their multiplicities.

3. Find the $y$-intercept by evaluating $f(0)$.

4. Use symmetry if applicable.

5. Plot additional points as needed.

2.4 Dividing Polynomials: Remainder and Factor Theorem

Long Division and Synthetic Division

• Long Division: Used to divide polynomials similarly to numerical long division.

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

Remainder Theorem

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

Factor Theorem

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

2.5 Zeros of Polynomial Functions

The Rational Zero Theorem

• Possible rational zeros are $\frac{\text{factors of the constant term}}{\text{factors of the leading coefficient}}$.

Properties of Roots

• A polynomial of degree $n$ has $n$ roots (counting multiplicities).

• Complex roots occur in conjugate pairs if coefficients are real.

The Fundamental Theorem of Algebra

• Every polynomial equation of degree $n \geq 1$ has at least one complex root.

The Linear Factorization Theorem

• Every polynomial of degree $n$ can be factored into $n$ linear factors (possibly with complex coefficients).

Descartes' Rule of Signs

• Gives the possible number of positive and negative real zeros by counting sign changes in $f(x)$ and $f(-x)$.

2.6 Rational Functions and Their Graphs

Definition of a Rational Function

• A rational function is a quotient of two polynomials: $f(x) = \frac{p(x)}{q(x)}$, $q(x) \neq 0$.

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

Vertical Asymptotes

• Vertical asymptotes occur at values of $x$ where $q(x) = 0$ and $p(x)$ and $q(x)$ have no common factors.

Horizontal Asymptotes

• If degree of numerator $<$ degree of denominator, horizontal asymptote is $y = 0$.

• If degrees are equal, horizontal asymptote is $y = \frac{a_n}{b_m}$ (ratio of leading coefficients).

• If degree of numerator $>$ degree of denominator, no horizontal asymptote.

Basic Reciprocal Functions

• $f(x) = \frac{1}{x}$ is an odd function with origin symmetry.

• $f(x) = \frac{1}{x^2}$ is an even function with $y$-axis symmetry.

Graphing Rational Functions

1. Determine symmetry.

2. Find $y$-intercept by evaluating $f(0)$.

3. Find $x$-intercepts by solving $p(x) = 0$.

4. Find vertical asymptotes by solving $q(x) = 0$.

5. Find horizontal asymptotes using degree rules.

6. Plot points and sketch the graph, considering asymptotes and intercepts.

2.7 Polynomial and Rational Inequalities

Definition

• A polynomial/rational inequality is an inequality involving a polynomial or rational function, such as $f(x) < 0$, $f(x) > 0$, $\frac{p(x)}{q(x)} \leq 0$.

Procedure for Solving Polynomial Inequalities

1. Express the inequality in the form $f(x) < 0$, $f(x) > 0$, etc.

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

3. Plot boundary points on a number line and divide into intervals.

4. Test a value from each interval in $f(x)$ to determine where the inequality holds.

5. Write the solution set, including or excluding boundary points as appropriate.

Example Table: Arrow Notation for Rational Functions

Symbol	Meaning
$x \to a^+$	$x$ approaches $a$ from the right
$x \to a^-$	$x$ approaches $a$ from the left
$x \to \infty$	$x$ increases without bound
$x \to -\infty$	$x$ decreases without bound

Additional info:

• Examples and exercises are provided throughout the notes for practice, including graphing, finding zeros, and solving equations and inequalities.

• These notes cover key Precalculus topics: complex numbers, quadratic and polynomial functions, rational functions, and inequalities.