1. Introduction to 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.
2. Polynomial Functions
- Polynomials: Expressions involving sums of powers of variables with coefficients.
- Standard Form: where
- Degree of a Polynomial: The highest power of the variable (e.g., 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.
3. Rational Functions
- Definition: A ratio of two polynomials, where and are polynomials.
- Domain: The set of all real numbers except where the denominator 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.
4. Complex Numbers
- Definition: Numbers in the form is the imaginary unit ().
- Operations:
- Addition/Subtraction: Combine real and imaginary parts separately.
- Multiplication: Use the distributive property and
- Division: Multiply the numerator and denominator by the conjugate of the denominator.
- Polar Form: Representing complex numbers as or , where and
- De Moivre’s Theorem: , useful for finding powers and roots of complex numbers.
5. Exponential and Logarithmic Functions
- Exponential Functions: Functions of the form , where and .
- Properties:
- Properties:
- Logarithmic Functions: The inverse of exponential functions,
- Properties:
- Properties:
- Natural Logarithm: is the logarithm to the base (Euler’s number, approximately 2.718).
- Change of Base Formula:
6. Systems of Equations and Inequalities
- 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.
7. Matrices and Determinants
- 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: , where is the identity matrix.
- Cramer’s Rule: A method for solving linear systems using determinants.
8. Sequences and Series
- Sequences: An ordered list of numbers defined by a rule.
- Arithmetic Sequence: The difference between consecutive terms is constant.
- Geometric Sequence: The ratio between consecutive terms is constant.
- Series: The sum of the terms of a sequence.
- Arithmetic Series: Sum of an arithmetic sequence, .
- Geometric Series: Sum of a geometric sequence, for
- 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.
9. Binomial Theorem
- Statement: Describes the algebraic expansion of powers of a binomial. , where is a binomial coefficient.
- Application: Useful in expanding polynomials and in probability theory.
10. Conic Sections
- Definition: Curves obtained by intersecting a cone with a plane.
- Types:
- Circle:
- Ellipse:
- Parabola:
- Hyperbola:
- Circle:
- 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.
11. Vectors and Vector Spaces
- Vectors: Quantities with both magnitude and direction. Represented as
- Operations:
- Addition/Subtraction: Component-wise addition or subtraction.
- Scalar Multiplication: Multiplying each component by a scalar.
- Dot Product: 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.
12. Linear Transformations
- Definition: Functions that map vectors to vectors, preserving vector addition and scalar multiplication.
- Matrix Representation: Any linear transformation can be represented by a matrix , where
- Eigenvalues and Eigenvectors: For a matrix , if , then is an eigenvalue and is an eigenvector.
- Diagonalization: The process of finding a diagonal matrix similar to a given matrix , such that