Fundamental Concepts Exam

Your success in the core mathematics curriculum begins here!

Cadets enter the U.S. Military Academy (USMA) with different math backgrounds. To successfully begin work in their science, technology, engineering and mathematics (STEM) courses, incoming cadets must arrive with the knowledge of certain mathematical skills and concepts. The Fundamental Concepts Exam (FCE) is designed to test those skills. All cadets are required to pass the test and demonstrate mastery of its topics before moving on to higher math courses.

Required Mathematical Skills for Prospective Cadets

1. Number Sense and Algebraic Foundations
   1. Use symbols and operations to represent relationships
   2. Understand how quantities relate in real-world situations such as rates and averages
   3. Apply basic properties of numbers to simplify and reason through problems
2. Exponents and Radicals
   1. Rewrite expressions using different forms of powers and roots
   2. Simplify expressions involving exponents
   3. Recognize how exponent rules apply in different contexts
3. Algebraic Expressions
   1. Add, subtract, multiply, and divide expressions
   2. Simplify expressions by combining like terms and expanding
   3. Recognize structure to make expressions easier to work with
4. Factoring and Structure
   1. Break numbers and expressions into simpler factors
   2. Recognize when factoring helps solve problems
   3. Use factoring to identify key values or solutions
5. Equations and Inequalities
   1. Solve equations involving one or more variables
   2. Solve and interpret inequalities
   3. Use multiple methods to find and verify solutions
6. Polynomials and Rational Expressions
   1. Work with expressions involving multiple terms
   2. Solve inequalities involving polynomial or rational relationships
   3. Understand how restrictions affect solutions
7. Linear Relationships
   1. Describe relationships that change at a constant rate
   2. Interpret and construct linear models
   3. Compare relationships that are parallel or perpendicular
8. Functions
   1. Understand how one quantity depends on another
   2. Identify inputs, outputs, and reasonable values
   3. Evaluate and combine functions in different ways
9. Quadratic and Nonlinear Relationships
   1. Solve problems involving curved or non-constant relationships
   2. Recognize key features such as turning points and intersections
   3. Compare linear and nonlinear behavior
10. Trigonometry
    1. Understand relationships between angles and sides of triangles
    2. Work with common angle measures and reference triangles
    3. Recognize patterns and key relationships among trigonometric functions
11. Exponential and Logarithmic Relationships
    1. Understand growth and decay processes
    2. Recognize the relationship between exponential and logarithmic ideas
    3. Simplify and solve problems involving these relationships
12. Graphing and Visualization
    1. Represent relationships using graphs
    2. Identify key features such as intercepts and general shape
    3. Understand how transformations affect graphs
13. Geometry and Measurement
    1. Apply concepts of distance, area, and volume
    2. Understand properties of common shapes
    3. Use geometric reasoning to solve real-world problems