Factoring and Quadratic Equations

Factoring Techniques

Factoring is a fundamental algebraic skill used to simplify expressions and solve equations. It involves rewriting a polynomial as a product of simpler polynomials.

• FOIL Method: Used for multiplying two binomials. FOIL stands for First, Outside, Inside, Last.

• Box Method: A visual method for multiplying polynomials by arranging terms in a grid.

Quadratic Equations

• Quadratic Formula: Used to solve equations of the form . Formula:

• Square Root Property: If , then .

Zeros of Polynomial Functions

• Factoring: Set the polynomial equal to zero and factor to find solutions.

• Long Division: Used to divide polynomials, especially when factoring is difficult.

• Synthetic Division: A shortcut for dividing a polynomial by a linear factor of the form .

Example:

Solve by factoring:

• Factor:

• Solutions: ,

Linear Equations and Inequalities

Solving Linear Equations

Linear equations have the form . Solving for involves isolating the variable.

• Example:

Graphing Inequalities

• Number Line: Shade the region representing the solution set.

• Cartesian Plane: For inequalities in two variables, shade the appropriate region on the graph.

Domain and Range

• Domain: All possible input values () for a function.

• Range: All possible output values () for a function.

Intercepts

• x-intercept: Set and solve for .

• y-intercept: Set and solve for .

Functions and Their Properties

Graphing Functions

• Linear:

• Quadratic:

• Inequality:

• Rational:

Symmetry Tests

• y-axis Symmetry: Replace with ; if unchanged, symmetric about y-axis.

• x-axis Symmetry: Replace with ; if unchanged, symmetric about x-axis.

• Origin Symmetry: Replace with and with ; if unchanged, symmetric about the origin.

Multiplicity of Zeros

• Odd Multiplicity: Graph crosses the x-axis.

• Even Multiplicity: Graph touches and turns at the x-axis.

Leading Coefficient and End Behavior

• Positive Leading Coefficient: Right end rises.

• Negative Leading Coefficient: Right end falls.

Transformations

• Shift: Moves the graph horizontally or vertically.

• Shrink/Stretch: Changes the steepness of the graph.

• Reflect: Flips the graph over an axis.

Vertex of Quadratics

• For , vertex at

Limits and Asymptotes

• Vertical Asymptote: Set denominator to zero in rational functions.

• Horizontal Asymptote: Compare degrees of numerator and denominator.

Logarithmic and Exponential Functions

• Exponential:

• Logarithmic:

• Conversion:

• Growth & Decay:

Piecewise Functions

• Defined by different expressions for different intervals of the domain.

• Example:

Linear Systems, Matrices, and Determinants

Solving Linear Systems

• Substitution Method: Solve one equation for a variable and substitute into the other.

• Addition (Elimination) Method: Add or subtract equations to eliminate a variable.

• Gaussian Elimination: Use row operations to reduce a system to row-echelon form.

Matrices

• Matrix Addition: Add corresponding elements.

• Matrix Multiplication: Multiply rows by columns.

• Scalar Multiplication: Multiply every entry by a constant.

• Solving Systems: Use matrices to represent and solve systems of equations.

Determinants and Cramer's Rule

• Determinant: For a matrix , determinant is .

• Cramer's Rule: Solve using determinants:

Sequences and Series

Arithmetic Sequences

• General Term:

• Common Difference:

Geometric Sequences

• General Term:

• Common Ratio:

Summation Notation

• Sum of first terms of arithmetic sequence:

• Sum of first terms of geometric sequence: (for )

Finding Specific Terms

• To find , use the general formula for the sequence.

Writing Formulas

• Identify and (arithmetic) or (geometric) to write the explicit formula.