Review of College Algebra: Foundations for Trigonometry

Review of College Algebra

Number Systems

The number system forms the basis of algebraic operations and includes several classifications:

• Real Numbers: Any number on the number line, including positive, negative, zero, fractions, and decimals.

• Integers: Positive and negative whole numbers, including zero.

• Prime Numbers: Numbers greater than 1 with only two positive divisors: 1 and itself.

Properties of Operations

Algebraic operations follow specific properties that simplify calculations:

• Commutative Property: Addition: $a + b = b + a$ Multiplication: $a \times b = b \times a$

• Associative Property: Addition: $(a + b) + c = a + (b + c)$ Multiplication: $(a \times b) \times c = a \times (b \times c)$

• Identity Property: Addition: $a + 0 = a$ Multiplication: $a \times 1 = a$

Order of Operations

To evaluate expressions correctly, follow the order of operations:

• Parentheses

• Exponents

• Multiplication and Division (left to right)

• Addition and Subtraction (left to right)

Prime Factorization

Prime factorization expresses a number as a product of prime numbers. This is useful for simplifying fractions and finding least common multiples.

• Example: $40 = 2 \times 2 \times 2 \times 5$

Fractions

Fractions represent a part of a whole and are fundamental in algebraic manipulation.

• Proper Fraction: Numerator is less than denominator.

• Improper Fraction: Numerator is greater than or equal to denominator.

• Mixed Number: Combination of a whole number and a fraction.

Adding and Subtracting Fractions

To add or subtract fractions, follow these steps:

1. Find the least common denominator (LCD).

2. Convert each fraction to an equivalent fraction with the LCD.

3. Add or subtract the numerators, keeping the denominator the same.

Multiplying and Dividing Fractions

Multiplication and division of fractions involve:

• Multiplication: Multiply numerators and denominators directly. $\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}$

• Division: Multiply by the reciprocal of the divisor. $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$

Exponents

Exponents indicate repeated multiplication of a number. Several rules govern their manipulation:

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

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

• Power Rule: $(a^m)^n = a^{m \times n}$

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

• Negative Exponent: $a^{-n} = \frac{1}{a^n}$

Radicals

Radicals represent roots of numbers. The principal root is the positive root for even indices.

• Product Rule: $\sqrt{a} \times \sqrt{b} = \sqrt{ab}$

• Quotient Rule: $\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$

• Simplifying Radicals: Factor the radicand and extract perfect squares.

Logarithms

Logarithms are the inverse of exponents and are used to solve equations involving exponentials.

• Definition: $\log_b(x) = n$ means $b^n = x$

• Product Rule: $\log_b(xy) = \log_b(x) + \log_b(y)$

• Quotient Rule: $\log_b(\frac{x}{y}) = \log_b(x) - \log_b(y)$

• Power Rule: $\log_b(x^n) = n \log_b(x)$

• Change of Base Rule: $\log_b(x) = \frac{\log_k(x)}{\log_k(b)}$

Decimals

Decimals are another way to represent fractions and are used in arithmetic operations:

• Multiplying Decimals: Multiply as whole numbers, then place the decimal point according to the total number of decimal places.

• Adding/Subtracting Decimals: Line up decimal points before performing the operation.

Ratios and Proportions

Ratios compare two quantities, while proportions set two ratios equal to each other.

• Ratio: $\frac{a}{b}$ or $a:b$

• Proportion: $\frac{a}{b} = \frac{c}{d}$

Scientific Notation

Scientific notation expresses very large or small numbers in the form $a \times 10^n$.

• Move the decimal point to create a number between 1 and 10, then multiply by a power of 10.

• Example: $5,000 = 5 \times 10^3$

Arithmetic Sequences

An arithmetic sequence is a list of numbers with a constant difference between consecutive terms.

• General Term: $a_n = a_1 + (n-1)d$

• Sum of Sequence: $S_n = \frac{n}{2}(a_1 + a_n)$

Prime Numbers

Prime numbers are fundamental in factorization and number theory.

• List of primes between 60 and 70: 61, 67

Examples and Applications

• Finding LCD for fractions

• Prime factorization for simplifying radicals

• Solving proportions in real-world contexts

• Expressing numbers in scientific notation

• Calculating terms and sums in arithmetic sequences