Fundamental Concepts of Algebra

Simplifying Expressions

Algebraic simplification involves reducing expressions to their simplest form using arithmetic operations and properties of exponents and radicals.

• Example: Simplify .

• Key Point: Rationalizing denominators is often required for final answers.

• Formula:

Factoring

Factoring is the process of expressing a polynomial as a product of its factors.

• Key Point: Common techniques include factoring trinomials, difference of squares, and sum/difference of cubes.

• Example: Factor as .

• Formula:

Functions and Graphs

Domain and Range

The domain of a function is the set of all possible input values, while the range is the set of all possible output values.

• Example: For , , .

Transformations of Functions

Transformations include translations, reflections, stretches, and compressions of graphs.

• Key Point: shifts the graph units right and units up.

• Example: Graph by reflecting, stretching, and translating the parent function .

Polynomial and Rational Functions

Multiplying and Simplifying Polynomials

Use distributive property and combine like terms to multiply polynomials.

• Example:

Dividing Rational Expressions

To divide rational expressions, multiply by the reciprocal of the divisor.

• Formula:

• Example:

Systems of Equations and Matrices

Solving Systems of Equations

Systems can be solved by substitution, elimination, or using matrices.

• Example: Solve , .

Augmented Matrices

An augmented matrix represents a system of linear equations for solution via row operations.

Conic Sections and Analytic Geometry

Circles

The equation of a circle in standard form is .

• Center:

• Radius:

• Example: has center and radius $3$.

Midpoint Formula

The midpoint of a segment with endpoints and is:

• Formula:

Sequences, Induction, and Probability

Arithmetic Sequences

An arithmetic sequence has a constant difference between consecutive terms.

• Formula:

Additional Topics

Word Problems and Applications

Precalculus often includes real-world applications such as cost analysis, projectile motion, and geometric problems.

• Example: Comparing internet service plans using systems of equations.

• Example: Modeling the height of a dropped object:

Graphing and Slope

The slope of a line perpendicular to is .

• Formula:

Summary Table: Key Algebraic Formulas