College Algebra Assignment Guide and Topic Overview

Overview of College Algebra Assignments and Topics

This guide outlines the structure of assignments, key textbook sections, and major topics covered in a College Algebra course. It is designed to help students organize their study and practice effectively for exams and assessments.

Assignment Structure and Recommendations

• Textbook: College Algebra with Intermediate Algebra: A Blended Course, by Beecher, Penna, Johnson, and Bittinger.

• Written Assignments: Complete and submit assigned textbook exercises, showing all work.

• Online Assignments: Complete MyLab Math assignments for additional practice and support.

• Suggested Practice: Additional exercises are recommended for extra practice but are not required.

• Assessment: Problems similar to homework may appear on quizzes and exams. Each chapter includes review sections and tests for further practice.

Unit 1: Functions, Systems of Equations, Polynomials, and Rational Expressions

Functions and Their Properties

Functions are fundamental objects in algebra, representing relationships between sets of inputs and outputs. Understanding their domains, ranges, and how to express these in interval notation is essential.

• Definition: A function is a relation in which each input has exactly one output.

• Domain: The set of all possible input values (x-values).

• Range: The set of all possible output values (y-values).

• Interval Notation: A concise way to describe sets of numbers, especially domains and ranges.

• Example: For $f(x) = \sqrt{x-2}$, the domain is $[2, \infty)$.

Systems of Equations

Systems of equations involve finding values that satisfy multiple equations simultaneously. Methods include substitution, elimination, and graphical solutions.

• Linear System: A set of two or more linear equations.

• Solution: The point(s) where all equations intersect.

• Example: Solve $\begin{cases} x + y = 5 \\ 2x - y = 1 \end{cases}$.

Polynomials and Polynomial Functions

Polynomials are expressions consisting of variables and coefficients, using only addition, subtraction, multiplication, and non-negative integer exponents.

• Standard Form: $a_nx^n + a_{n-1}x^{n-1} + \dots + a_1x + a_0$

• Degree: The highest exponent of the variable.

• Example: $f(x) = 2x^3 - 5x^2 + x - 7$ is a cubic polynomial.

Rational Expressions

Rational expressions are ratios of polynomials. Simplifying, finding domains, and performing operations are key skills.

• Definition: An expression of the form $\frac{P(x)}{Q(x)}$, where $Q(x) \neq 0$.

• Domain: All real numbers except where $Q(x) = 0$.

• Example: $\frac{x^2 - 1}{x - 1}$ simplifies to $x + 1$ for $x \neq 1$.

Unit 2: Complex Rational Expressions, Rational Equations, and Radical Expressions

Complex Rational Expressions

These are rational expressions where the numerator, denominator, or both contain rational expressions themselves.

• Simplification: Multiply numerator and denominator by the least common denominator (LCD) to simplify.

• Example: $\frac{\frac{1}{x} + \frac{2}{y}}{\frac{3}{x}}$

Rational Equations and Applications

Rational equations are equations involving rational expressions. Solving them often requires finding a common denominator.

• Steps:

  1. Find the LCD of all denominators.

  2. Multiply both sides by the LCD to clear denominators.

  3. Solve the resulting equation.

  4. Check for extraneous solutions.

• Example: Solve $\frac{1}{x} + \frac{2}{x+1} = 3$.

Radical Expressions and Functions

Radical expressions involve roots, such as square roots or cube roots. Understanding their domains and how to simplify them is crucial.

• Definition: An expression containing a root, such as $\sqrt{x}$ or $\sqrt[3]{x}$.

• Domain: For even roots, the radicand must be non-negative.

• Example: The domain of $f(x) = \sqrt{4 - 3x}$ is $x \leq \frac{4}{3}$.

Unit 3: Division of Radical Expressions, Radical Equations, Quadratic and Polynomial Functions

Division of Radical Expressions

Dividing radical expressions often involves rationalizing denominators and simplifying.

• Rationalizing: Multiply numerator and denominator by a suitable radical to eliminate radicals from the denominator.

• Example: $\frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}$

Radical Equations and Applications

Equations involving radicals can be solved by isolating the radical and raising both sides to a power.

• Steps:

  1. Isolate the radical expression.

  2. Raise both sides to the appropriate power.

  3. Solve the resulting equation.

  4. Check for extraneous solutions.

• Example: Solve $\sqrt{x+3} = x - 1$.

Quadratic Functions and Equations

Quadratic equations are second-degree polynomials. They can be solved by factoring, completing the square, or using the quadratic formula.

• Standard Form: $ax^2 + bx + c = 0$

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

• Discriminant: $D = b^2 - 4ac$ determines the number of real solutions.

• Example: Solve $2x^2 - 8x + 3 = 0$ by completing the square.

Polynomial Functions

Polynomial functions of higher degree can be analyzed for zeros, end behavior, and graphing.

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

• Example: Find the zeros of $f(x) = x^3 - 6x^2 + 11x - 6$.

Unit 4: Rational Functions, Polynomial and Rational Inequalities

Rational Functions

Rational functions are quotients of polynomials. Key features include vertical and horizontal asymptotes, intercepts, and domain.

• Vertical Asymptotes: Occur where the denominator is zero and the numerator is nonzero.

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

• Example: For $f(x) = \frac{2x}{x^2 - 1}$, vertical asymptotes at $x = 1, -1$.

Polynomial and Rational Inequalities

Solving inequalities involving polynomials or rational expressions requires finding critical points and testing intervals.

• Steps:

  1. Set the inequality to zero.

  2. Find zeros of numerator and denominator.

  3. Use a sign chart or number line to test intervals.

  4. Express the solution in interval notation.

• Example: Solve $\frac{x^2 - 1}{x^2 + 5x + 6} \leq 0$.

Sample Table: Assignment Organization

The following table summarizes how assignments are organized by section, page, and type:

Section	Page	Submitted Problems	Suggested Problems
§2.2	92	17, 18, 56, 58	1, 3, 5, 7, 23, 52, 54
§3.2	163	15, 17, 21
§4.2	235	23, 55, 69, 79, 87	27, 63
§5.3	314	11, 15, 19, 33 (use long division)
§6.1	374	23, 27, 45, 49, 71, 75, 79	7, 19, 21, 51, 59, 67
§7.4 pt 1	477	Extra: Solve by completing the square: $2x^2-8x+3=0$, $2y^2+12y+3=0$	1, 3, 21, 23, 37, 49
§8.5	561	3, 5, 8, 10 (vertical and horizontal asymptotes only)

Key Exam Dates

• Test 1: Sections 2.2, 2.3, 3.2, 3.3, 4.2, 4.7, 5.1, 5.2, 5.3

• Test 2: Sections 5.4, 5.5, 5.6, 6.1 - 6.4

• Test 3: Sections 6.5 - 6.8, 7.4, 7.5, 8.1

• Final Exam: Cumulative, scheduled for Wednesday, December 9, 3:30 pm-5:30 pm

Additional Info

• Imaginary numbers (square roots of negative numbers) are mentioned but are beyond the scope of this course.

• Students are encouraged to use all available resources, including MyLab Math and textbook reviews, for comprehensive preparation.