Problem 20 - A Computational Introduction to Number Theory and Algebra

Question

Let \(g \in R[X],\) and let \(x \in R\) be a root of \(g .\) Show that \(x\) is a multiple root of \(g\) if and only if \(x\) is also a root of \(\mathbf{D}(g)\) (see Exercise 7.18 ).

Step-by-Step Solution

Answer

Question: Prove that \(x\) is a multiple root of \(g\) if and only if \(x\) is also a root of \(\mathbf{D}(g)\), where \(\mathbf{D}(g)\) is the first derivative of the polynomial \(g\). Solution: We can prove this by showing that 1. If \(x\) is a root of \(g\) and \(\mathbf{D}(g)\), then it's a multiple root, and 2. If \(x\) is a multiple root of \(g\), then it's a root of \(\mathbf{D}(g)\). If \(x\) is a root of \(g\) and \(\mathbf{D}(g)\), it implies that the highest power term of \(g(X)\) has a factor of \((X-x)^n\), meaning \(x\) is a multiple root of \(g\). On the other hand, if \(x\) is a multiple root of \(g\), then \(g(X) = (X-x)^k h(X)\), where \(k \geq 2\) and \(h\) is a polynomial. When calculating \(\mathbf{D}(g)\), we find that \(\mathbf{D}(g)(x) = 0\), meaning \(x\) is a root of \(\mathbf{D}(g)\). Thus, \(x\) is a multiple root of \(g\) if and only if \(x\) is a root of \(\mathbf{D}(g)\).

Step 1: Part 1: If \(x\) is a root of \(g\) and \(\mathbf{D}(g)\), then it's a multiple root

Assume \(x\) is a root of \(g\) and \(\mathbf{D}(g)\). Now, let \(g(X) = a_n(X-x)^n + a_{n-1}(X-x)^{n-1} + \cdots + a_1(X-x) + a_0\) for a polynomial \(g \in R[X]\). Since \(x\) is a root of \(g\), \(g(x)=0\). Now, we find the derivative of \(g\), using the power rule for derivatives: \(\mathbf{D}(g) = na_n(X-x)^{n-1} + (n-1)a_{n-1}(X-x)^{n-2} + \cdots + a_1\) \(x\) being a root of \(\mathbf{D}(g)\) implies that \(\mathbf{D}(g)(x) = 0\). Thus, \(na_n(x-x)^{n-1} + (n-1)a_{n-1}(x-x)^{n-2} + \cdots + a_1 = 0\) Notice that the first term of the derivative has a factor of \((X-x)^{n-1}\) since the highest power term of \(g(X)\) was raised to \(n\). In other words, \(g(X)\) has a factor of \((X-x)^n\) if and only if \(\mathbf{D}(g)(x)=0\). Therefore, \(x\) is a multiple root of \(g\).

Step 2: Part 2: If \(x\) is a multiple root of \(g\), then it's a root of \(\mathbf{D}(g)\)

Assume \(x\) is a multiple root of \(g\). This means there exists an integer \(k \geq 2\) such that \(g(X) = (X-x)^k h(X)\) for some polynomial \(h \in R[X]\). Now, let's find the derivative of \(g\) using the product rule for derivatives: \(\mathbf{D}(g) = \mathbf{D}((X-x)^k h(X)) = (X-x)^k \mathbf{D}(h) + k(X-x)^{k-1}h(X)\) Now, evaluate \(\mathbf{D}(g)\) at \(x\): \(\mathbf{D}(g)(x) = (x-x)^k \mathbf{D}(h(x)) + k(x-x)^{k-1}h(x) = 0\) Thus, \(x\) is a root of \(\mathbf{D}(g)\). Therefore, we conclude that \(x\) is a multiple root of \(g\) if and only if \(x\) is a root of \(\mathbf{D}(g)\).

Key Concepts

Derivative

A derivative is a concept that helps us understand how a function changes at any given point. Think of it as a way to measure how fast something is growing or shrinking. When dealing with polynomials, derivatives are particularly useful because they allow us to explore the nature of the roots of the polynomial. For a polynomial function, its derivative is another polynomial of a lower degree.

• For example, if you have a polynomial like \( f(x) = x^3 - 3x + 2 \), taking the derivative would give you \( f'(x) = 3x^2 - 3 \).
• This new function \( f'(x) \) provides information about where \( f(x) \) increases or decreases and can identify potential multiple roots.

When we say that a number \( x \) is a root of a polynomial, it means that the polynomial equals zero when \( x \) is substituted into it. However, if \( x \) is also a root of the polynomial's derivative, it indicates that the root might appear more than once, making it a multiple root.

Polynomial Roots

Understanding the roots of a polynomial is essential in fields like engineering, physics, and computer science. A root of a polynomial is simply a value for which the polynomial equals zero. But what about multiple roots? Multiple roots, also known as repeated roots, occur when a root appears more than once in the polynomial.

• Consider the polynomial \( g(x) = (x - 2)^3 \). Here, \( x = 2 \) is a root that appears three times, making it a multiple root.
• This means that if you set \( g(x) = 0 \), the solution \( x = 2 \) satisfies the equation three times.

Multiple roots often impact the behavior of the graph of a polynomial. For instance, in the above example, the graph would just touch the x-axis at \( x = 2 \) but never cross it. Detecting such roots is critical in various mathematical applications and can be done using derivatives.

Power Rule for Derivatives

The power rule for derivatives is a straightforward yet powerful concept when dealing with polynomials. It states that if you have a term like \( x^n \), its derivative is \( nx^{n-1} \). This rule helps simplify the process of finding derivatives, especially for functions that are sums of power terms.

• For instance, for the term \( 4x^5 \), the derivative is \( 20x^4 \).
• Applying the power rule repeatedly allows you to find the derivative of even complex polynomials easily.

The power rule is especially crucial when determining the nature of polynomial roots. By easily finding the derivative of a polynomial, one can verify whether a given root is a multiple root. Since multiple roots will also be roots of the derivative, using the power rule effectively helps identify these intricate relationships within polynomials.