1.1 Review of Algebra

Algebra provides the foundation for much of the work you'll do in calculus. This section will review the essential algebraic concepts and techniques you need to be comfortable with before advancing further. These include operations with real numbers, manipulation of algebraic expressions, solving equations and inequalities, and an introduction to functions.

1.1.1 Real Numbers and Their Properties

The set of real numbers, denoted by \( \mathbb{R} \), includes all rational and irrational numbers. The key properties of real numbers are:

1. Commutative Property:
   • Addition: \( a + b = b + a \)
   • Multiplication: \( ab = ba \)
2. Associative Property:
   • Addition: \( (a + b) + c = a + (b + c) \)
   • Multiplication: \( (ab)c = a(bc) \)
3. Distributive Property:
   \[
   a(b + c) = ab + ac
   \]
4. Additive Inverse:
   For any real number \( a \), there exists an additive inverse \( -a \) such that:
   \[
   a + (-a) = 0
   \]
5. Multiplicative Inverse:
   For any real number \( a \neq 0 \), there exists a multiplicative inverse \( \frac{1}{a} \) such that:
   \[
   a \cdot \frac{1}{a} = 1
   \]

1.1.2 Operations with Algebraic Expressions

Algebraic expressions involve variables and constants combined through addition, subtraction, multiplication, division, and exponentiation. Some common operations are:

• Addition/Subtraction: Combine like terms (terms that have the same variables raised to the same powers).
  \[
  3x + 5x = 8x
  \]
  \[
  7x^2 - 3x^2 = 4x^2
  \]
• Multiplication: Apply the distributive property when necessary.
  \[
  (2x + 3)(x - 4) = 2x(x - 4) + 3(x - 4) = 2x^2 - 8x + 3x - 12 = 2x^2 - 5x - 12
  \]
• Division: When dividing polynomials, simplify terms if possible. For example:
  \[
  \frac{6x^2}{3x} = 2x
  \]

1.1.3 Factoring Algebraic Expressions

Factoring is the process of expressing a polynomial as a product of its factors. Some common factoring techniques include:

1. Factoring out the Greatest Common Factor (GCF):
   \[
   6x^2 + 9x = 3x(2x + 3)
   \]
2. Factoring Trinomials:
   Consider a quadratic expression \( ax^2 + bx + c \). To factor, find two numbers that multiply to \( ac \) and add to \( b \).
   \[
   x^2 + 5x + 6 = (x + 2)(x + 3)
   \]
3. Difference of Squares:
   \[
   a^2 - b^2 = (a - b)(a + b)
   \]
   Example:
   \[
   x^2 - 9 = (x - 3)(x + 3)
   \]

1.1.4 Solving Equations and Inequalities

Solving Linear Equations

A linear equation has the form \( ax + b = 0 \). To solve for \( x \), isolate the variable by applying inverse operations.

Example:
\[
2x - 4 = 10 \quad \Rightarrow \quad 2x = 14 \quad \Rightarrow \quad x = 7
\]

Solving Quadratic Equations

Quadratic equations have the form \( ax^2 + bx + c = 0 \). Methods for solving include:

1. Factoring:
   \[
   x^2 + 5x + 6 = 0 \quad \Rightarrow \quad (x + 2)(x + 3) = 0
   \]
   Solutions: \( x = -2 \) or \( x = -3 \)
2. Quadratic Formula:
   For any quadratic equation \( ax^2 + bx + c = 0 \), the solution is given by:
   \[
   x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
   \]
   Example:
   Solve \( 2x^2 - 4x - 6 = 0 \) using the quadratic formula.
   \[
   x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(2)(-6)}}{2(2)} = \frac{4 \pm \sqrt{16 + 48}}{4} = \frac{4 \pm \sqrt{64}}{4} = \frac{4 \pm 8}{4}
   \]
   Solutions: \( x = 3 \) or \( x = -\frac{1}{2} \)

Solving Inequalities

To solve an inequality, perform operations similar to those used in solving equations, but remember that multiplying or dividing both sides by a negative number reverses the inequality sign.

Example:
\[
3x - 7 > 2 \quad \Rightarrow \quad 3x > 9 \quad \Rightarrow \quad x > 3
\]

1.1.5 Exponents and Logarithms

Exponent Rules

Exponents follow these basic rules:

1. \( a^m \cdot a^n = a^{m+n} \)
2. \( \frac{a^m}{a^n} = a^{m-n} \)
3. \( (a^m)^n = a^{mn} \)
4. \( a^{-n} = \frac{1}{a^n} \)
5. \( a^0 = 1 ) (for ( a \neq 0 )\)

Example:
\[
2^3 \cdot 2^2 = 2^{3+2} = 2^5 = 32
\]

Logarithms

The logarithm is the inverse operation of exponentiation. For \( a > 0 \) and \( a \neq 1 \), the logarithm base \( a \) of \( x \) is the number \( y \) such that:
\[
a^y = x \quad \text{or} \quad \log_a{x} = y
\]

Some useful properties of logarithms:

1. \( \log_a{(xy)} = \log_a{x} + \log_a{y} \)
2. \( \log_a{\left(\frac{x}{y}\right)} = \log_a{x} - \log_a{y} \)
3. \( \log_a{x^n} = n \log_a{x} \)
4. \( \log_a{a} = 1 ), and ( \log_a{1} = 0 \)

Example:
\[
\log_2{32} = 5 \quad \text{because} \quad 2^5 = 32
\]

1.1.6 Introduction to Functions

A function is a relation between a set of inputs (domain) and a set of possible outputs (range) where each input is related to exactly one output. A function is typically written as ( f(x) ), where ( x ) is the input variable.

Example of a function:
\[
f(x) = 2x + 3
\]
For \( x = 1 \), \( f(1) = 2(1) + 3 = 5 \).

Key Concepts:

• Domain: The set of all possible input values for the function.
• Range: The set of all possible output values.
• Vertical Line Test: A graph represents a function if and only if no vertical line intersects the graph at more than one point.

This completes the review of basic algebra concepts essential for understanding calculus. Each of these topics will play a critical role in the manipulation of functions, solving equations, and other techniques necessary as we progress into more advanced topics in calculus.