Foundations and Problem Solving in College Algebra: Key Concepts and Practice

Numbers and Their Classifications

Types of Numbers

Understanding the different types of numbers is fundamental in algebra. Numbers are classified based on their properties and uses.

• Natural Numbers: Counting numbers starting from 1 (e.g., 1, 2, 3, ...).

• Whole Numbers: Natural numbers plus zero (e.g., 0, 1, 2, ...).

• Integers: Whole numbers and their negatives (e.g., -3, -2, -1, 0, 1, 2, ...).

• Rational Numbers: Numbers that can be expressed as a fraction $\frac{a}{b}$, where $a$ and $b$ are integers and $b \neq 0$.

• Irrational Numbers: Numbers that cannot be written as a simple fraction (e.g., $\sqrt{2}$, $\pi$).

• Real Numbers: All rational and irrational numbers.

Example: $-8$ is an integer, rational, and real number.

Operations with Fractions and Decimals

Simplifying Fractions

To simplify a fraction, divide both the numerator and denominator by their greatest common divisor (GCD).

• Example: $\frac{12}{36} = \frac{1}{3}$

Adding, Subtracting, Multiplying, and Dividing Fractions

• Add/Subtract: Find a common denominator, then add/subtract numerators.

• Multiply: Multiply numerators and denominators directly.

• Divide: Multiply by the reciprocal of the divisor.

Example: $\frac{7}{11} \times \frac{11}{7} = 1$

Order of Operations

PEMDAS/BODMAS

Follow the order: Parentheses, Exponents, Multiplication/Division, Addition/Subtraction.

• Example: $2 \times (3 + 4) = 2 \times 7 = 14$

Exponents and Powers

Exponent Rules

Exponents indicate repeated multiplication. Key rules include:

• Product Rule: $a^m \times a^n = a^{m+n}$

• Quotient Rule: $\frac{a^m}{a^n} = a^{m-n}$

• Power Rule: $(a^m)^n = a^{mn}$

• Zero Exponent: $a^0 = 1$ (for $a \neq 0$)

Example: $(x^2)^3 = x^{2 \times 3} = x^6$

Polynomials and Their Classification

Types of Polynomials

Polynomials are algebraic expressions with one or more terms, each consisting of a variable raised to a non-negative integer power.

• Monomial: One term (e.g., $5x$)

• Binomial: Two terms (e.g., $x + 2$)

• Trinomial: Three terms (e.g., $x^2 + 3x + 2$)

Degree of a Polynomial: The highest exponent of the variable in the polynomial.

Example: $x^3 + 2x^2 + x$ has degree 3.

Solving Equations and Problem Solving

Linear Equations

Linear equations are equations of the first degree, meaning the variable is not raised to any power other than one.

• General Form: $ax + b = c$

• Solving: Isolate the variable using inverse operations.

Example: $x + 8 = 14 \implies x = 6$

Word Problems and Algebraic Expressions

Translating verbal statements into algebraic expressions is a key skill.

• Example: "Five times the sum of a number and 4" is $5(x + 4)$

Properties of Operations

Distributive Property

The distributive property allows you to multiply a sum by multiplying each addend separately and then adding the products.

• Formula: $a(b + c) = ab + ac$

• Example: $3(x + 2) = 3x + 6$

Evaluating and Simplifying Expressions

Combining Like Terms

Like terms have the same variable raised to the same power. Combine them by adding or subtracting their coefficients.

• Example: $2x + 3x = 5x$

Numerical Coefficient

The numerical coefficient is the number multiplying the variable in a term.

• Example: In $-8x$, the coefficient is -8.

Opposites and Reciprocals

Additive Inverse (Opposite)

The additive inverse of a number is the number that, when added to the original, yields zero.

• Example: The opposite of $-8$ is $8$.

Reciprocal

The reciprocal of a nonzero number $a$ is $\frac{1}{a}$.

• Example: The reciprocal of $\frac{2}{3}$ is $\frac{3}{2}$.

Application and Problem Solving

Translating and Solving Word Problems

Many algebra problems require translating real-world situations into mathematical equations.

• Example: "The number of graduate students is 20,000 fewer than the number of undergraduate students." If $x$ is the number of undergraduates, then graduates = $x - 20,000$.

Tables: Number Classifications

Number	Natural	Whole	Integer	Rational	Irrational	Real
5	✔	✔	✔	✔		✔
-8			✔	✔		✔
$\sqrt{2}$					✔	✔

Additional info:

• Some questions involve basic graphing and representation, which are foundational for later graphing topics.

• Problems include both computational and conceptual elements, suitable for exam preparation.