BackCollege Algebra: Core Concepts, Equations, and Applications
Study Guide - Smart Notes
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Radicals and Exponents
Evaluating and Simplifying Radical Expressions
Radical expressions involve roots, such as square roots or cube roots. Simplifying these expressions often requires factoring and rationalizing denominators.
Key Point: The principal square root of a number is its non-negative root. For example, is the non-negative number whose square is .
Key Point: To simplify , factor each radicand and combine like terms:
Example:
Combine:
Additional info: Always express answers in simplest radical form unless otherwise specified.
Evaluating Exponential Expressions
Exponential expressions involve repeated multiplication of a base. Negative and fractional exponents represent roots and reciprocals.
Key Point:
Example:
Solving Equations
Linear Equations
Linear equations are equations of the first degree, meaning the variable is not raised to any power other than one.
Key Point: To solve , divide both sides by 3: .
Quadratic Equations
Quadratic equations are equations of the form . They can be solved by factoring, completing the square, or using the quadratic formula.
Quadratic Formula:
Example: For , , , :
Discriminant:
Solutions: , so or
Additional info: If the discriminant is negative, there are no real solutions.
Equations Involving Radicals
To solve equations involving square roots, isolate the radical and square both sides.
Example:
Example:
Factoring and Polynomials
Factoring Polynomials
Factoring is the process of writing a polynomial as a product of its factors.
Key Point: To factor , look for two numbers that multiply to and add to .
Example:
Multiplying Polynomials
Use the distributive property (FOIL for binomials) to multiply polynomials.
Example:
Rational Expressions and Complex Fractions
Simplifying Rational Expressions
Rational expressions are fractions where the numerator and/or denominator are polynomials. Simplify by factoring and reducing common factors.
Example:
Factor denominators: ,
Find common denominator and combine.
Complex Fractions
Complex fractions have fractions in the numerator, denominator, or both. Simplify by finding a common denominator.
Example:
Rewrite denominator:
So,
Functions and Applications
Function Notation and Applications
Functions describe relationships between variables. Applications often involve interpreting or solving for variables in context.
Example: The annual revenue of a company is given by .
To find when , set and solve for .
Solving for a Variable
To solve for a variable in terms of others, isolate the desired variable using algebraic operations.
Example:
Rearrange:
Inequalities and Number Lines
Graphing Solutions on a Number Line
Inequalities can be represented graphically on a number line, showing all possible solutions.
Example: is shown as an arrow starting at 1 and extending to the right.
Additional Topics
Solving for y in Terms of x
Some equations require solving for one variable in terms of another, often involving radicals.
Example:
To solve for , rearrange and solve the resulting quadratic equation.
Word Problems and Applications
Word problems require translating real-world scenarios into algebraic equations and solving for unknowns.
Example: Health care expenditures , find when .
Solve for .
