College Algebra Final Exam Study Guide

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 $ax^2 + bx + c = 0$. Formula: $x = \frac{ -b \pm \sqrt{b^2 - 4ac} }{2a}$

• Square Root Property: If $x^2 = k$, then $x = \pm \sqrt{k}$.

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 $(x - c)$.

Example:

Solve $x^2 - 5x + 6 = 0$ by factoring:

• Factor: $(x - 2)(x - 3) = 0$

• Solutions: $x = 2$, $x = 3$

Linear Equations and Inequalities

Solving Linear Equations

Linear equations have the form $ax + b = c$. Solving for $x$ involves isolating the variable.

• Example: $2x + 3 = 7 \implies x = 2$

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 ($x$) for a function.

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

Intercepts

• x-intercept: Set $y = 0$ and solve for $x$.

• y-intercept: Set $x = 0$ and solve for $y$.

Functions and Their Properties

Graphing Functions

• Linear: $y = x + 1$

• Quadratic: $f(x) = x^2 + 10$

• Inequality: $5x^2 > 20$

• Rational: $f(x) = \frac{x - 1}{x^2 - 4}$

Symmetry Tests

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

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

• Origin Symmetry: Replace $x$ with $-x$ and $y$ with $-y$; 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 $f(x) = ax^2 + bx + c$, vertex at $x = -\frac{b}{2a}$

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: $y = a^x$

• Logarithmic: $y = \log_a x$

• Conversion: $y = \log_a x \iff a^y = x$

• Growth & Decay: $A = A_0 e^{kt}$

Piecewise Functions

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

• Example: $f(x) = \begin{cases} x^2, & x < 0 \\ x + 1, & x \geq 0 \end{cases}$

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 $2 \times 2$ matrix $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$, determinant is $ad - bc$.

• Cramer's Rule: Solve $Ax = b$ using determinants: $x_i = \frac{\det(A_i)}{\det(A)}$

Sequences and Series

Arithmetic Sequences

• General Term: $a_n = a_1 + (n-1)d$

• Common Difference: $d = a_{n} - a_{n-1}$

Geometric Sequences

• General Term: $a_n = a_1 r^{n-1}$

• Common Ratio: $r = \frac{a_n}{a_{n-1}}$

Summation Notation

• Sum of first $n$ terms of arithmetic sequence: $S_n = \frac{n}{2}(a_1 + a_n)$

• Sum of first $n$ terms of geometric sequence: $S_n = a_1 \frac{1 - r^n}{1 - r}$ (for $r \neq 1$)

Finding Specific Terms

• To find $a_{28}$, use the general formula for the sequence.

Writing Formulas

• Identify $a_1$ and $d$ (arithmetic) or $r$ (geometric) to write the explicit formula.