Question

Let \(R\) be a ring, let \(g \in R[X],\) with \(\operatorname{deg}(g)=k \geq 0,\) and let \(x\) be an element of \(R\). Show that: (a) there exist an integer \(m,\) with \(0 \leq m \leq k,\) and a polynomial \(q \in R[X],\) such that $$ g=(X-x)^{m} q \text { and } q(x) \neq 0 $$ and moreover, the values of \(m\) and \(q\) are uniquely determined; (b) if we evaluate \(g\) at \(X+x,\) we have $$ g(X+x)=\sum_{i=0}^{k} b_{i} X^{i} $$ where \(b_{0}=\cdots=b_{m-1}=0\) and \(b_{m}=q(x) \neq 0\)

Short answer

Question: Given a polynomial \(g(X)\) of degree \(k \geq 0\), and a real number \(x\), show that there exist unique non-negative integers \(m\) and a polynomial \(q(X)\) with \(q(x) \neq 0\) such that \(g(X) = (X-x)^m q(X)\) and \(g(X+x) = \sum_{i=0}^{k} b_{i} X^{i}\) with \(b_0 = \cdots = b_{m-1} = 0\) and \(b_m = q(x) \neq 0\). Solution: The existence and uniqueness of \(m\) and \(q\) can be established by using the division algorithm for polynomials. With a proper choice of \(m\) and \(q\), we can ensure that \(g(X) = (X-x)^m q(X)\) and \(q(x) \neq 0\). Evaluating \(g(X+x)\), it is found that the coefficients of \(g(X+x)\) satisfy the given conditions.

Step-by-step solution

Existence of m and q

By the division algorithm, we can write \(g(X) = (X-x)^m q(X) + r(X)\) where the degree of \(r\) is strictly less than the degree of \((X-x)^m\). We must show that we can pick \(m\) and \(q\) so that \(r(x) = 0\) and \(q(x) \neq 0\). Let's pick \(m\) to be the smallest integer such that \(r(x) = 0\). This means that \(g(X) = (X-x)^m q(X)\) where \(q(X)\) is a quotient polynomial. Since \(r(x) = 0\), \(q(x) \neq 0\).

Uniqueness of m and q

Suppose there is another pair \(m'\) and \(q'\) such that \(g(X) = (X-x)^{m'} q'(X)\) and \(q'(x) \neq 0\). Without loss of generality, we assume that \(m' \geq m\). By dividing \((X-x)^{m'} q'(X)\) by \((X-x)^m\), we obtain a remainder polynomial $ )r'(X)\( of degree less than \)m\(. However, since we already know that \)(X-x)^m\( divides \)g(X)\(, the remainder \) )r'(X) = (X-x)^{m'-m}q'(X)\( must also be zero. This forces \)m'=m\( and \)q'(X)=q(X)$, proving uniqueness.

Evaluating g(X+x)

Now, we need to evaluate \(g(X + x)\). We apply the substitution \(X \mapsto X + x\) into the polynomial \(g(X)\): $$ g(X+x) = [(X+x)-x]^m q(X+x) = X^m q(X+x) $$

Coefficients of g(X+x)

Since the degree of \(g(X+x)\) is \(k \geq 0\), let's write it as $$ g(X+x) = \sum_{i=0}^k {b_i}{X^i} $$ Notice that for \(0 \leq i \leq m-1\), all the coefficients \(b_i\) from \(X^i\) terms are zero. This is because in \(g(X+x) = X^m q(X+x)\), the smallest power of \(X\) that can appear is \(X^m\). Hence, \(b_0 = \cdots = b_{m-1} = 0\). Also, the coefficient \(b_m\) comes from the term \(X^m\) in \(g(X+x) = X^m q(X+x)\), which is equal to \(q(x)\), as given. Thus, we have shown that $$ g(X+x) = \sum_{i=0}^{k} b_{i} X^{i} $$ with \(b_0 = \cdots = b_{m-1} = 0\) and \(b_m = q(x) \neq 0\).

Key concepts

Ring Theory

In mathematics, especially in abstract algebra, a ring is a fundamental structure that extends the concepts of addition and multiplication. A ring consists of a set equipped with two binary operations: addition and multiplication, satisfying specific conditions. Unlike fields, not every element in a ring has a multiplicative inverse.

Rings are pivotal in solving polynomial equations, as seen in operations involving ring elements like polynomials. Important properties of a ring include:

• Addition: The operation is associative and commutative, with an identity element (often zero).
• Multiplication: Associative with an identity element, though not necessarily commutative or invertible for every element.
• Distributive Law: Multiplication distributes over addition, forming the basis of the ring's structure.

Understanding rings and their properties allows us to manage equations and expressions within algebraic systems efficiently. They provide the framework needed to explore and solve polynomial equations analytically.

Polynomial Evaluation

Polynomial evaluation refers to determining the value of a polynomial function at a specific point or for a given value of the variable. This concept plays an essential role in algebra, particularly when solving equations or simplifying expressions.

To evaluate a polynomial, plug the value into the expression and perform the necessary arithmetic operations. Consider a polynomial function:

• For a polynomial \( p(X) = a_n X^n + a_{n-1} X^{n-1} + \cdots + a_0 \), to evaluate at \( X = x \), substitute \( x \) for \( X \) in the polynomial.
• The result is \( p(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_0 \).

Polynomial evaluation is not just about substitution; it also involves understanding the structure and degree of the polynomial. This knowledge aids in polynomial division and determining how alterations, such as shifts, affect polynomial expressions. In polynomial division, this becomes crucial when using the Division Algorithm to break down polynomials into simpler components.

Algorithmic Mathematics

Algorithmic mathematics involves using systematic procedures (algorithms) to solve mathematical problems. This systematic approach allows for precise, repeatable solutions, especially in polynomial operations.

When working with polynomials, algorithms such as the Euclidean algorithm play a significant role in division processes, like when dividing polynomials. The Division Algorithm offers a method to express a polynomial \( g(X) \) as \( g(X) = (X-x)^m q(X) \), where \( q(X) \) is another polynomial and the remainder is zero or lower degree.

Steps involved include:

• Identifying the degree of the polynomial involved and determining the terms of interest.
• Systematically substituting values and simplifying expressions to verify the division results.
• Ensuring all terms fulfill the initial conditions, like the uniqueness of division results and evaluation accuracy.

These algorithms not only simplify finding solutions but also ensure that the expressions conform to the mathematical rules governing polynomial structures. Understanding and applying algorithmic mathematics is crucial in managing complex polynomial expressions, leading to efficient problem-solving in algebra.