Polynomial Magic

Combinatorics · Axiom Academy

Discover how polynomials secretly count combinations for you. You have three coins worth 1¢, 5¢, and 10¢. For each coin, you can choose to take it or leave it. Click coins to select or deselect them. Try different combinations! Each coin gives us a choice: take it (x value ) or leave it (x 0 = 1). Let's write this as a polynomial! The complete polynomial for all coins: Let's multiply out the polynomial and see what happens. Watch how each term appears! Now let's solve a classic problem: How many ways can you make 10¢ using unlimited pennies, nickels, and dimes? Polynomials encode combinatorial choices. Each factor represents an independent decision, and multiplication combines all possibilities. The coefficient of x n automatically counts the number of ways to achieve a sum of n. No manual enumeration needed! This technique is called "generating functions" - one of the most powerful tools in combinatorics. You've just discovered the foundation!

This is the written version of the interactive lesson above. See the full Combinatorics course.