IndietroCollege Algebra Final Review: Key Concepts and Practice
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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 Step-by-step: So, 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: Common denominator is 15:
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: Combine like terms: Subtract 85: Divide by -7:
Solving Equations with Fractions
Key Point: Multiply both sides by the least common denominator to clear fractions before solving.
Example: Multiply both sides by 5:
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 is in slope-intercept form, where the slope is 2 and the y-intercept is 1.
Example: For , calculate and plot the points , , .
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 and :
Parallel:
Perpendicular:
Example: and are perpendicular because .
Exponents and Polynomials
Exponent Rules
Exponent rules help simplify expressions involving powers.
Product Rule:
Quotient Rule:
Power Rule:
Negative Exponent:
Example:
Polynomials: Multiplication and Factoring
Multiplying polynomials uses the distributive property (FOIL for binomials). Factoring reverses this process.
Key Point: To multiply , use FOIL: .
Example: First, Then,
Factoring Example:
Functions
Linear Functions and Applications
Linear functions have the form and are used to model real-world relationships.
Key Point: The slope represents the rate of change, and is the initial value.
Example: The cost of renting a storage space for months:
Rational Expressions
Simplifying Rational Expressions
To simplify rational expressions, factor numerators and denominators and reduce common factors.
Example:
Exponential & Scientific Notation
Scientific Notation
Scientific notation expresses numbers as , where and is an integer.
Standard Form:
Expanded Form:
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: is an equation; is an expression.
Summary Table: Exponent Rules
Rule | Formula | Example |
|---|---|---|
Product of Powers | ||
Quotient of Powers | ||
Power of a Power | ||
Negative Exponent |
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.
