Problem 10 - A History of Mathematics: An Introduction

Question

Complete al-Samaw'al's procedure of dividing \(20 x^{2}+30 x\) by \(6 x^{2}+12\) to get the result stated in the text. Prove that the coefficients of the quotient satisfy the rule \(a_{n+2}=-2 a_{n}\) where \(a_{n}\) is the coefficient of \(\frac{1}{n}\)

Step-by-Step Solution

Answer

Based on the given division problem, we found the quotient expression to be \(Q(x) = \frac{10}{3} - \frac{5}{6x}\). When we tried to verify the rule \(a_{n+2} = -2a_n\) for the non-zero coefficients, we found that it did not hold for both n=0 and n=1. Therefore, the coefficients of the quotient do not satisfy the given rule.

Step 1: Divide the given polynomials using al-Samaw'al's procedure

First, we'll set up the division like this: \(\frac{20x^2 + 30x}{6x^2 + 12}\) To proceed with al-Samaw'al's procedure, divide the first term of the numerator by the first term of the denominator: \(\frac{20x^2}{6x^2} = \frac{10}{3}\) Next, multiply the entire denominator by the result: \(\frac{10}{3}(6x^2 + 12) = 20x^2 + 40x.\) Subtract this new expression from the original numerator: \((20x^2 + 30x) - (20x^2 + 40x) = -10x.\) Now, divide the result by the same \(6x^2\) we used before: \(\frac{-10x}{6x^{2}} = -\frac{5}{6x}\) Now, our expression for the quotient is: \(\frac{10}{3} - \frac{5}{6x}\)

Step 2: Write the result of the division as a series

We can view the quotient expression as a series, so: \(Q(x) = \frac{10}{3} - \frac{5}{6x}\) As a series, we have: \(Q(x) = a_0 + a_1x + a_2x^2 + ...\) where \(a_0 = \frac{10}{3}\), \(a_1 = -\frac{5}{6}\), and the rest of the coefficients are zero, since there are no other terms in the quotient.

Step 3: Verify that the coefficients of the quotient satisfy the given rule \(a_{n+2} = -2a_n\)

To verify that the coefficients of the quotient \(Q(x)\) satisfy the rule \(a_{n+2} = -2a_n\), let's check the rule for the non-zero coefficients: For \(n = 0\): \(a_{0} = \frac{10}{3}\), \(a_{2} = 0\) \(a_{n+2} = -2a_{n} \iff a_2 = -2a_0 \iff 0 = -2 \cdot \frac{10}{3} \iff 0 = -\frac{20}{3}\) This is not true, so the rule doesn't hold for n=0. For \(n = 1\): \(a_{1} = -\frac{5}{6}\), \(a_{3} = 0\) \(a_{n+2} = -2a_{n} \iff a_3 = -2a_1 \iff 0 = -2 \cdot -\frac{5}{6} \iff 0 = \frac{10}{6}\) This is also not true, so the rule doesn't hold for n=1. Thus, the coefficients of the quotient do not satisfy the rule \(a_{n+2} = -2a_n\).

Key Concepts

al-Samaw'al's procedure

Al-Samaw'al's procedure is a method of polynomial division named after the medieval mathematician al-Samaw'al. It simplifies polynomial division, providing a straightforward approach. To understand it, think of it like long division but for polynomials. This method is especially useful in ensuring that each step of the division accounts for the powers of the variables involved. When dividing the polynomials \(20x^2 + 30x\) by \(6x^2 + 12\), the process begins by focusing on the leading terms.

• First, divide the first term of the numerator \(20x^2\) by the first term of the denominator \(6x^2\).
• This gives the coefficient \(\frac{10}{3}\).
• Next, multiply the entire denominator by this coefficient and subtract from the original numerator.
• Continue this process until no further division can be made without resulting in a fraction that includes the variable as the denominator.

Al-Samaw'al's procedure allows us to derive a quotient of polynomials that are part integer, part fractional powers of variables. This gives a clearer understanding of polynomial behavior.

Coefficient Rule

In polynomial division, coefficients of the resulting quotient often follow specific patterns or rules. One such rule proposed in this exercise is that the coefficients \(a_{n+2}\) of the quotient should satisfy \(-2a_n\). However, upon closer examination through the division of the given polynomials, it turns out that this rule does not apply in this case. Let’s break it down.

• The quotient derived was \(\frac{10}{3} - \frac{5}{6x}\); hence coefficients are \(a_0 = \frac{10}{3}\), \(a_1 = -\frac{5}{6}\), and all other higher-degree coefficients are zero.
• Testing this against the rule, we see that \(a_2\) should be equal to \(-2\cdot a_0\) which does not hold true as \(a_2 = 0\).
• Similarly, \(a_3\) should be \(-2\cdot a_1\) but this is also not satisfied as \(a_3 = 0\).

This discrepancy highlights the importance of verifying such rules, as they may not universally apply or require specific polynomial conditions to hold.

Mathematical Proof

Mathematical proof is a logical argument demonstrating the truth of a mathematical statement. In polynomial division, attempting to prove coefficient rules requires careful verification. Let's use the given problem as an example to illustrate the proof process.Attempting to apply and prove the rule \(a_{n+2} = -2a_n\) involves evaluating the coefficients from the derived polynomial quotient \(Q(x) = \frac{10}{3} - \frac{5}{6x}\). Here's the step-by-step reasoning:

• Start by identifying the coefficients. For \(a_0 = \frac{10}{3}\) and \(a_1 = -\frac{5}{6}\), assess whether \(a_2 = a_3 = 0\) works with the rule.
• For \(n = 0\), calculate \(a_2 = -2 \times a_0\) which fails since \(a_2 = 0\) does not equal \(-\frac{20}{3}\).
• Similarly, for \(n = 1\), \(a_3\) should equal \(-2 \times a_1\), but it also does not satisfy the equation as \(a_3 = 0\) is not \(\frac{10}{6}\).

Such proofs necessitate methodical checking of every step. They ensure that any derived rules or patterns accurately reflect the relationships between polynomial coefficients. Even when a rule appears mathematically sound, empirical verification, like the one in this exercise, clarifies applicability and correctness.