# Advanced Compound Interest: Mathematical Origins & Research

Before reading this analysis, it is advised to read the __Regular Investing analysis__, as it develops the basis behind many of our foundational formulas represented here.

We know that any polynomial can be represented as a finite, descending sum via __Polynomial Reduction Theory__:

*f *(x) = L * Σ { xᵃ(a!) / xⁿ(a-n)! }, where the summation is bounded by {n* *=0, a}

As an example, let's display f (x) = 3x³ + 12x² + 14 as a finite sum:

*f *(x) = L * Σ { x³(3!) / xⁿ(3-n)! }, where the summation is bounded by {n* *=0, 3}

### = L * Σ { 6x³ / xⁿ(3-n)! }

### = 6x³ L * Σ { 1 / xⁿ(3-n)! }

### = 6x³ L * (⅙ + 1/(2x) + 1/x² + 1/x³)

### = x³ L + 3x²L + 6xL + 6L

### (3x³ + 12x² + 14) / (x³ + x² + 6x + 6) = L

Essentially, we're drawing out the constants of the equation so that only the base values remain, so that when we take a derivative, the base values (the variables) are the only things that change. This makes it extremely easy to do otherwise-impossible calculations in the span of minutes.

Let's simplify L down.

### (3x³ + 12x² + 14) / (x³ + x² + 6x + 6) = L

### L = 5x + (5/2)x² + ½x³ + 12/x + 14/x³ + 14/x² + 7/(3x) + 52/3

### From the regular investing analysis, we also learned that investment interest could be modeled (via compound interest) as

### I = Prᵗ + Mr ᵗ ( r ᵗ-1 ) / ( r -1 )

T will always end up as a constant, no matter how the formula is manipulated. Therefore, the compound interest itself can be viewed as a polynomial: all we need to do is apply PRT to it, so we can find the optimal strategy for our situation.