College Algebra Final Review: Key Concepts and Practice

Review of Algebra

Order of Operations

Order of operations is essential for simplifying mathematical expressions correctly. The standard order is Parentheses, Exponents, Multiplication/Division (left to right), Addition/Subtraction (left to right), often abbreviated as PEMDAS.

• Key Point: Always perform calculations inside parentheses first, then exponents, followed by multiplication/division, and finally addition/subtraction.

• Example: Simplify $-12 + 6^2 - (6 - 4)^2 - 6 \div 3^2 - 6 + 12$ Step-by-step: $6^2 = 36$ $(6-4)^2 = 2^2 = 4$ $3^2 = 9$ $6 \div 9 = 2/3$ So, $-12 + 36 - 4 - 2/3 - 6 + 12$ Combine like terms to get the final answer.

Fractions and Simplification

Combining and simplifying fractions requires a common denominator.

• Key Point: To add or subtract fractions, rewrite each with a common denominator, then combine numerators.

• Example: $\frac{3}{5} - \frac{7}{3} + \frac{2}{15}$ Common denominator is 15: $\frac{9}{15} - \frac{35}{15} + \frac{2}{15} = \frac{9 - 35 + 2}{15} = \frac{-24}{15} = -\frac{8}{5}$

Equations & Inequalities

Solving Linear Equations

Linear equations can be solved by isolating the variable using inverse operations.

• Key Point: Combine like terms and use addition, subtraction, multiplication, or division to isolate the variable.

• Example: $85 - 2x - 5x = 17$ Combine like terms: $85 - 7x = 17$ Subtract 85: $-7x = 17 - 85 = -68$ Divide by -7: $x = \frac{-68}{-7} = \frac{68}{7}$

Solving Equations with Fractions

• Key Point: Multiply both sides by the least common denominator to clear fractions before solving.

• Example: $\frac{6w}{5} - \frac{2}{5} = 4w$ Multiply both sides by 5: $6w - 2 = 20w$ $-2 = 14w$ $w = -\frac{2}{14} = -\frac{1}{7}$

Graphs of Equations

Graphing Linear Equations

To graph a linear equation, create a table of values, plot the points, and draw the line through them.

• Key Point: The equation $y = 2x + 1$ is in slope-intercept form, where the slope is 2 and the y-intercept is 1.

• Example: For $x = -2, 0, 2$, calculate $y$ and plot the points $(-2, -3)$, $(0, 1)$, $(2, 5)$.

Parallel and Perpendicular Lines

Lines are parallel if they have the same slope and perpendicular if the product of their slopes is -1.

• Key Point: For lines $y = m_1x + b_1$ and $y = m_2x + b_2$:

  • Parallel: $m_1 = m_2$

  • Perpendicular: $m_1 \cdot m_2 = -1$

• Example: $y = 4x - 7$ and $y = -\frac{1}{4}x - 1$ are perpendicular because $4 \times -\frac{1}{4} = -1$.

Exponents and Polynomials

Exponent Rules

Exponent rules help simplify expressions involving powers.

• Product Rule: $x^a \cdot x^b = x^{a+b}$

• Quotient Rule: $\frac{x^a}{x^b} = x^{a-b}$

• Power Rule: $(x^a)^b = x^{ab}$

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

• Example: $\frac{x^{12}}{x^7} = x^{12-7} = x^5$

Polynomials: Multiplication and Factoring

Multiplying polynomials uses the distributive property (FOIL for binomials). Factoring reverses this process.

• Key Point: To multiply $(a+b)(c+d)$, use FOIL: $ac + ad + bc + bd$.

• Example: $(x+4)^3 = (x+4)(x+4)(x+4)$ First, $(x+4)(x+4) = x^2 + 8x + 16$ Then, $(x^2 + 8x + 16)(x+4) = x^3 + 12x^2 + 48x + 64$

• Factoring Example: $x^2 - 3x - 4 = (x-4)(x+1)$

Functions

Linear Functions and Applications

Linear functions have the form $y = mx + b$ and are used to model real-world relationships.

• Key Point: The slope $m$ represents the rate of change, and $b$ is the initial value.

• Example: The cost $y$ of renting a storage space for $x$ months: $y = 78x + 100$

Rational Expressions

Simplifying Rational Expressions

To simplify rational expressions, factor numerators and denominators and reduce common factors.

• Example: $\frac{12x^2}{18x} = \frac{2x}{3}$

Exponential & Scientific Notation

Scientific Notation

Scientific notation expresses numbers as $a \times 10^n$, where $1 \leq a < 10$ and $n$ is an integer.

• Standard Form: $3.2 \times 10^4$

• Expanded Form: $32,000$

Equations vs. Expressions

Identifying Equations and Expressions

An equation contains an equals sign and shows a relationship between two expressions. An expression does not have an equals sign.

• Example: $4x - 2 = 10$ is an equation; $4x - 2$ is an expression.

Summary Table: Exponent Rules

Rule	Formula	Example
Product of Powers	$x^a \cdot x^b = x^{a+b}$	$x^2 \cdot x^3 = x^5$
Quotient of Powers	$\frac{x^a}{x^b} = x^{a-b}$	$\frac{x^5}{x^2} = x^3$
Power of a Power	$(x^a)^b = x^{ab}$	$(x^2)^3 = x^6$
Negative Exponent	$x^{-a} = \frac{1}{x^a}$	$x^{-2} = \frac{1}{x^2}$

Additional info:

• Some context and explanations have been expanded for clarity and completeness.

• Examples are based on the problems shown in the images and standard College Algebra curriculum.