Advanced Algebra

Definition: Advanced Algebra builds on the foundational concepts of Elementary Algebra, focusing on more complex equations, functions, and systems. It is essential for understanding higher-level mathematics like calculus, linear algebra, and abstract algebra.

Polynomials: Expressions involving sums of powers of variables with coefficients.
- Standard Form: $a_nx^n + a_{n-1}x^{n-1} + \dots + a_1x + a_0$ where $a_{n} \neq 0$
- Degree of a Polynomial: The highest power of the variable (e.g., $3x^4 + 2x^3 + x + 5$ has a degree of 4).
Operations:
- Addition/Subtraction: Combine like terms.
- Multiplication: Distribute each term.
- Division: Long division or synthetic division.
Factoring: Breaking down polynomials into products of lower-degree polynomials.
- Methods: Common factor, grouping, difference of squares, sum/difference of cubes, and quadratic trinomials.

Definition: A ratio of two polynomials, $f(x) = \frac{P(x)}{Q(x)}$ where $P(x)$ and $Q(x)$ are polynomials.
Domain: The set of all real numbers except where the denominator $Q(x)$ is zero.
Asymptotes:
- Vertical Asymptote: Occurs where the denominator is zero.
- Horizontal Asymptote: Determined by the degrees of the polynomials in the numerator and denominator.
- Oblique Asymptote: Occurs when the degree of the numerator is exactly one more than the degree of the denominator.

Definition: Numbers in the form $a + ib, where$ $i$ is the imaginary unit ( $i^2 = -1$ ).
Operations:
- Addition/Subtraction: Combine real and imaginary parts separately.
- Multiplication: Use the distributive property and $i^2 = -1$
- Division: Multiply the numerator and denominator by the conjugate of the denominator.
Polar Form: Representing complex numbers as $r(\cos \theta + i \sin \theta)$ or $r e^{i\theta}$ , where $r = \sqrt{a^2 + b^2}$ and $θ = \tan^{- 1} (\frac{b}{a})$
De Moivre’s Theorem: $[r(\cos \theta + i \sin \theta)]^n = r^n (\cos n\theta + i \sin n\theta)$ , useful for finding powers and roots of complex numbers.

Exponential Functions: Functions of the form $f(x) = a \cdot b^x$ , where $b > 0$ and $b \neq 1$ .
- Properties:
  - $b^m \times b^n = b^{m+n}$
  - $\frac{b^m}{b^n} = b^{m-n}$
  - $(b^{m})^{n} = b^{m n}$
Logarithmic Functions: The inverse of exponential functions, $y = \log_{b} (x) means$ $b^y = x$
- Properties:
  - $\log_b(xy) = \log_b(x) + \log_b(y)$
  - $\log_{b} (\frac{x}{y}) = \log_{b} (x) - \log_{b} (y)$
  - $\log_b(x^y) = y \log_b(x)$
Natural Logarithm: $\ln(x)$ is the logarithm to the base $e$ (Euler’s number, approximately 2.718).
Change of Base Formula: $\log_{b} (x) = \frac{\log_{c} (x)}{\log_{c} (b)}$

Linear Systems:
- Solving Methods: Substitution, elimination, matrix methods (e.g., Gaussian elimination), and graphical methods.
- Number of Solutions: Consistent and independent (one solution), consistent and dependent (infinitely many solutions), or inconsistent (no solution).
Nonlinear Systems:
- Include equations like circles, parabolas, and ellipses.
- Solving Methods: Substitution, elimination, and graphical methods.
Systems of Inequalities:
- Graphical Solutions: Graph each inequality and find the intersection (solution region).
- Linear Programming: Optimizing a linear objective function subject to a system of linear inequalities.

Matrix Operations:
- Addition/Subtraction: Add or subtract corresponding elements.
- Multiplication: The dot product of rows and columns.
- Determinants: A scalar value that can be computed from the elements of a square matrix, useful for solving systems of linear equations.
  - Properties: If the determinant is zero, the matrix is singular and has no inverse.
The inverse of a Matrix: $A^{- 1 is the matrix that satisfies}$ $A \cdot A^{-1} = I$ , where $I$ is the identity matrix.
Cramer’s Rule: A method for solving linear systems using determinants.

Sequences: An ordered list of numbers defined by a rule.
- Arithmetic Sequence: The difference between consecutive terms is constant. $a_{n} = a_{1} + (n - 1) d$
- Geometric Sequence: The ratio between consecutive terms is constant. $a_n = a_1 \cdot r^{n-1}$
Series: The sum of the terms of a sequence.
- Arithmetic Series: Sum of an arithmetic sequence, $S_n = \frac{n}{2}(a_1 + a_n)$ .
- Geometric Series: Sum of a geometric sequence, $S_n = a_1 \cdot \frac{1-r^n}{1-r}$ for $r \neq 1.$
Infinite Series: Sum of an infinite sequence. Converges if the sum approaches a finite limit.
- Convergence Tests: Tests like the ratio and root tests determine whether an infinite series converges or diverges.

Statement: Describes the algebraic expansion of powers of a binomial. $(x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k$ , where $\binom{n}{k}$ is a binomial coefficient.
Application: Useful in expanding polynomials and in probability theory.

Definition: Curves obtained by intersecting a cone with a plane.
Types:
- Circle: $x^2 + y^2 = r^2$
- Ellipse: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$
- Parabola: $y = a x^{2} + b x + c$
- Hyperbola: $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$
Focus-Directrix Property: Each conic section has a focus and a directrix, and the distance from any point on the curve to the focus is proportional to its distance to the directrix.

Vectors: Quantities with both magnitude and direction. Represented as $\mathbf{v} = \langle v_1, v_2, \dots, v_n \rangle$
Operations:
- Addition/Subtraction: Component-wise addition or subtraction.
- Scalar Multiplication: Multiplying each component by a scalar.
- Dot Product: $\mathbf{u} \cdot \mathbf{v} = u_1v_1 + u_2v_2 + \dots + u_nv_n$ gives a scalar.
- Cross Product: Only defined in three dimensions, gives a vector perpendicular to the plane containing the original vectors.
Vector Spaces: A set of vectors closed under addition and scalar multiplication, with a defined zero vector.

Definition: Functions that map vectors to vectors, preserving vector addition and scalar multiplication.
Matrix Representation: Any linear transformation can be represented by a matrix $A$ , where $T(\mathbf{v}) = A\mathbf{v}$
Eigenvalues and Eigenvectors: For a matrix $A$ , if $A\mathbf{v} = \lambda\mathbf{v}$ , then $\lambda$ is an eigenvalue and $\mathbf{v}$ is an eigenvector.
Diagonalization: The process of finding a diagonal matrix $D$ similar to a given matrix $A$ , such that $A = PDP^{-1}$