Problem 10 - A History of Mathematics: An Introduction

Question

Maestro Dardi gave a rule to solve the fourth-degree equation \(x^{4}+b x^{3}+c x^{2}+d x=e\) as \(x=\sqrt[4]{(d / b)^{2}+e}-\) \(\sqrt{d / b}\). His problem illustrating the rule is the following: A man lent 100 lire to another and after 4 years received back 160 lire for principal and (annually compounded) interest. What is the interest rate? As in the text's example, set \(x\) as the monthly interest rate in denarii per lira. Show that this problem leads to the equation \(x^{4}+80 x^{3}+2400 x^{2}+32,000 x=96,000\) and that the solution found by "completing the fourth power" is given by the stated formula.

Step-by-Step Solution

Answer

Answer: The monthly interest rate is given by the expression x = √⁴(256000) - √(400).

Step 1: Translate the given problem into a polynomial equation

Let \(x\) be the monthly interest rate in denarii per lira, then \((1 + x)^{48}\) is the factor by which the initial loan amount will grow over the 4 years (since there are 48 months in 4 years). The total amount received after 4 years is 160 lire, which is composed of the initial loan amount 100 lire and the interest earned. Therefore, the equation can be written as: $$(1 + x)^{48} \cdot 100 = 100 + 160$$ $$(1 + x)^{48} = 2.6$$ Using the binomial theorem, we can expand the left side of the equation and keeping the terms up to x^4, we get: $$1 + 48x + \frac{48 \cdot 47}{2}x^2 + \frac{48 \cdot 47 \cdot 46}{3 \cdot 2}x^3 + \frac{48 \cdot 47 \cdot 46 \cdot 45}{4 \cdot 3 \cdot 2 }x^4 \approx 2.6$$ Subtract 1 from both sides to obtain the equation as follows: $$x^4 + 80x^3 + 2400x^2 + 32,000x = 96,000$$

Step 2: Apply Maestro Dardi's Rule for Fourth-Degree Equations

According to Maestro Dardi's rule, the solution of the fourth-degree equation \(x^4 + bx^3 + cx^2 + dx = e\) can be given as: $$x = \sqrt[4]{\left(\frac{d}{b}\right)^2 + e} - \sqrt{\frac{d}{b}}$$

Step 3: Substitute the Coefficients from the Given Equation

From the generated equation \(x^4 + 80x^3 + 2400x^2 + 32,000x = 96,000\), the coefficients are as follows: $$b = 80$$ $$d = 32,000$$ $$e = 96,000$$ Substitute these values into Maestro Dardi's Rule: $$x = \sqrt[4]{\left(\frac{32,000}{80}\right)^2 + 96,000} - \sqrt{\frac{32,000}{80}}$$

Step 4: Simplify and Solve the Equation

Simplify the expression for x: $$x = \sqrt[4]{\left(\frac{400}{1}\right)^2 + 96,000} - \sqrt{\frac{400}{1}}$$ Now, calculate each part separately: $$\sqrt[4]{\left(\frac{400}{1}\right)^2 + 96,000} = \sqrt[4]{160000 + 96000} = \sqrt[4]{256000}$$ Also, calculate the square root: $$\sqrt{\frac{32,000}{80}} = \sqrt{400}$$ And finally, solve for x: $$x = \sqrt[4]{256000} - \sqrt{400}$$ This expression gives the monthly interest rate in denarii per lira.

Key Concepts

Polynomial Equations

A polynomial equation is an equation formed by adding, subtracting, and multiplying constants and variables raised to a power. In this problem, we are dealing with a fourth-degree polynomial equation, which is an equation where the highest exponent of the variable, in this case, is 4.

In mathematical terms, the general form of a fourth-degree equation is: \[ x^4 + bx^3 + cx^2 + dx = e \] Each term is a part of the polynomial:

• \(x^4\) is the fourth-degree term.
• \(bx^3\) is the cubic term.
• \(cx^2\) is the quadratic term.
• \(dx\) is the linear term.
• \(e\) is the constant.

This kind of equation can be quite complex to solve, but by using mathematical rules and techniques, such as those provided by Maestro Dardi, we can find the solutions efficiently. These techniques involve advanced algebra and sometimes numerical methods to approximate the roots.

Binomial Theorem

The Binomial Theorem is a powerful tool in algebra for expanding expressions like \((1 + x)^{48}\). It allows us to express expanded polynomials easily without having to manually multiply the terms.

For any expression of the form \((a + b)^n\), the theorem states: \[ (a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k}b^k \] Where \(\binom{n}{k}\) is the binomial coefficient and can be calculated using factorials: \[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \] In the given exercise, the expression \((1 + x)^{48}\) is expanded to its fourth-degree term to simplify the equation into something manageable: \[ 1 + 48x + 1128x^2 + 17296x^3 + 194580x^4 \approx 2.6 \] This approximation helps in focusing on the most significant terms that influence the outcome when setting \((1 + x)^{48} \approx 2.6\). The theorem not only provides a way to organize polynomial computations but also bridges the understanding of how higher powers and terms interact in complex equations.

Interest Rate Calculation

Interest rate calculations are crucial in finance and investments, determining how much money will grow over time. In this example, we have a man lending money and receiving more back after a set period due to interest.

When we talk about compounded interest, the interest calculation becomes more intricate. Compounded interest means that interest is calculated on the initial principal and also on the accumulated interest from previous periods. For annually compounded interest over 4 years, the number of compounding periods is high. This is why we took the fourth degree polynomial approach in mathematics for modeling purposes.

The equation derived from this financial scenario, \(x^4 + 80x^3 + 2400x^2 + 32,000x = 96,000\), represents the entire growth factor of the loan. In this equation:

• The initial loan is 100 lire.
• The total amount after 4 years is 160 lire, making it a 60% increase.

This demonstrates how the mixture of mathematical concepts can tackle real-life financial problems, providing insights into loans, investments, and savings calculations. Understanding such interactions can deeply impact day-to-day financial decisions.