College Algebra (MATH 1030) Syllabus and Study Guide

Course Overview

Introduction

This course, College Algebra (MATH 1030), is a foundational mathematics course designed to prepare students for further study in mathematics and related fields. The syllabus outlines the main topics, course structure, assessment methods, and support resources. The content aligns closely with the standard precalculus curriculum, covering algebraic concepts, functions, equations, inequalities, matrices, and systems of equations.

Main Topics and Subtopics

Prerequisites: Fundamental Concepts of Algebra

• Radicals and Rational Exponents: Understanding expressions involving roots and fractional exponents. Example: Simplify $\sqrt[3]{x^4}$ and $x^{4/3}$.

• Factoring Polynomials: Techniques for breaking down polynomials into products of simpler polynomials. Example: Factor $x^2 - 5x + 6$ as $(x-2)(x-3)$.

Equations and Inequalities

• Linear Equations and Rational Equations: Solving equations of the form $ax + b = 0$ and equations involving rational expressions. Example: Solve $\frac{2x}{x-1} = 3$.

• Quadratic Equations: Solving equations of the form $ax^2 + bx + c = 0$ using factoring, completing the square, and the quadratic formula. Formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

• Complex Numbers: Introduction to numbers of the form $a + bi$, where $i^2 = -1$. Example: Solve $x^2 + 4 = 0$; solution: $x = 2i, x = -2i$.

• Linear and Absolute Value Inequalities: Solving inequalities and understanding solution sets. Example: Solve $|x - 3| < 5$.

Functions and Graphs

• Basics of Functions and Their Graphs: Definition of a function, domain, range, and graphical representation. Example: $f(x) = x^2$; domain: all real numbers, range: $[0, \infty)$.

• Linear Functions and Slope: Understanding the slope-intercept form $y = mx + b$ and calculating slope. Formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$

• Distance and Midpoint Formulas; Circles: Calculating the distance between two points and the midpoint. Distance Formula: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ Midpoint Formula: $M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$ Equation of a Circle: $(x - h)^2 + (y - k)^2 = r^2$

• Transformations of Functions: Shifts, stretches, and reflections of function graphs. Example: $f(x) = x^2$ shifted up by 3: $f(x) = x^2 + 3$.

• Combinations and Inverse Functions: Addition, subtraction, multiplication, division, and composition of functions; finding inverses. Example: If $f(x) = 2x + 1$, then $f^{-1}(x) = \frac{x - 1}{2}$.

Polynomial and Rational Functions

• Quadratic and Polynomial Functions: Analyzing graphs, zeros, and end behavior. Example: $f(x) = x^3 - 2x^2 + x$; find zeros and sketch graph.

• Dividing Polynomials; Remainder and Factor Theorems: Using synthetic and long division. Remainder Theorem: If $f(x)$ is divided by $x - a$, the remainder is $f(a)$.

• Zeros of Polynomial Functions: Finding roots and their multiplicities. Example: $f(x) = (x-1)^2(x+2)$; zeros: $x=1$ (multiplicity 2), $x=-2$.

• Rational Functions and Their Graphs: Analyzing asymptotes, holes, and behavior. Example: $f(x) = \frac{x+1}{x-2}$; vertical asymptote at $x=2$.

• Polynomial and Rational Inequalities: Solving inequalities involving polynomials and rational expressions.

Exponential and Logarithmic Functions

• Exponential Functions: Functions of the form $f(x) = a^x$. Example: $f(x) = 2^x$.

• Logarithmic Functions: Inverse of exponential functions; $y = \log_a x$ means $a^y = x$. Example: $\log_2 8 = 3$ because $2^3 = 8$.

• Properties of Logarithms: Laws such as $\log_a(xy) = \log_a x + \log_a y$, $\log_a(x^k) = k \log_a x$. Example: $\log_2(16) = \log_2(2^4) = 4 \log_2(2) = 4$.

• Exponential and Logarithmic Equations: Solving equations involving exponentials and logarithms. Example: Solve $2^x = 8$; $x = 3$.

Systems of Equations and Matrices

• Systems of Linear Equations in Two and Three Variables: Solving using substitution, elimination, and matrix methods. Example: Solve $\begin{cases} x + y = 3 \\ 2x - y = 0 \end{cases}$.

• Systems of Nonlinear Equations: Solving systems where at least one equation is nonlinear. Example: $\begin{cases} x^2 + y^2 = 25 \\ y = 2x \end{cases}$.

• Introduction to Matrices: Matrix notation, addition, multiplication, and using matrices to solve systems. Example: $\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} + \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix} = \begin{bmatrix} 6 & 8 \\ 10 & 12 \end{bmatrix}$

Assessment and Grading

Assessment Methods

• Homework: Weekly assignments submitted online; unlimited attempts, instant feedback, late penalty applies.

• Quizzes: Timed, online quizzes; two attempts, instant feedback.

• Tests: Four proctored tests, one per module; single attempt, no aids allowed.

• Final Exam: Comprehensive, proctored, single attempt, no aids allowed.

Grading Scale

Grade	GPA
A	4.0
A-	3.7
B+	3.3
B	3.0
B-	2.7
C+	2.3
C	2.0
C-	1.7
D+	1.3
D	1.0
D-	0.7
Ex	Excused
DL	Delayed
FN	Failure/non-participation

Weight Distribution

Assessment	Weight
Homeworks	12.5%
Quizzes	12.5%
Tests (4)	50% (12.5% each)
Final Exam	25%
Total	100%

Course Schedule (Topics by Week)

Week	Topics
1	P3: Radicals and Rational Exponents; P5: Factoring Polynomials; 1.2 Linear Equations and Rational Equations
2	1.3 Models and Applications; 1.4 Complex Numbers; 1.5 Quadratic Equations
3	1.6 Other Types of Equations; 1.7 Linear Inequalities
4	Review/Test 1
5	1.7 Absolute Value Inequalities; 2.1 Basics of Functions and Their Graphs; 2.2 More on Functions and Their Graphs
6	2.3 Linear Functions and Slope; 2.4 More on Slope; 2.8 Distance and Midpoint Formulas; Circles
7	2.5 Transformations of Functions; 2.6 Combinations of Functions; 2.7 Inverse Functions
8	Review/Test 2
9	3.1 Quadratic Functions; 3.2 Polynomial Functions and Their Graphs; 3.3 Dividing Polynomials; Remainder and Factor Theorems
10	3.4 Zeros of Polynomial Functions; 3.5 Rational Functions and Their Graphs; 3.6 Polynomial and Rational Inequalities
11	Review/Test 3
12	4.1 Exponential Functions; 4.2 Logarithmic Functions; 4.3 Properties of Logarithms; 4.4 Exponential and Logarithmic Equations
13	8.1 System of Linear Equations in 2 Variables; 8.2 Systems of Linear Equations in 3 Variables; 8.4 Systems of Nonlinear Equations in 2 Variables
14	9.1, 9.2 Introduction to Matrices; Review/Test 4
15	Test 4; Review for Final Exam; Practice Final Test

Student Support and Policies

• Participation: Regular engagement in assignments, quizzes, and discussions is required.

• Academic Integrity: Plagiarism and cheating are strictly prohibited and subject to university disciplinary action.

• Disability Access: Accommodations are available for students with documented needs.

• Technical Support: Assistance with Canvas and course technology is available.

• Tutoring: Free tutoring and Active Learning Assistant sessions are offered.

Summary

This syllabus provides a comprehensive overview of College Algebra, covering all major precalculus topics. Students are expected to master algebraic techniques, function analysis, equations, inequalities, and matrix operations. The course is structured with regular assessments and ample support resources to ensure student success.