Question

Let \(\alpha \in \mathbb{R}\) be a parameter and \(f, g_{a} \in \mathbb{R}[x]\) monic polynomials with res \(\left(f, g_{\alpha}\right)=\alpha^{3}+\alpha^{2}+\) \(\alpha+1\). Determine all values of \(\alpha\) for which \(\operatorname{ged}\left(f, g_{i x}\right) \neq 1\).

Short answer

\(\alpha = -1, i, -i\)

Step-by-step solution

Understand the Given Problem

We need to determine all values of \( \alpha \) for which the greatest common divisor (gcd) of two monic polynomials \( f \) and \( g_{ix} \) is not 1. Additionally, we're given that the resultant of \( f \) and \( g_{\alpha} \) is \( \alpha^3 + \alpha^2 + \alpha + 1 \). This means we will first understand the conditions under which the gcd is not 1.

Basic Theory Review

The gcd of two polynomials is not 1 if they have a common factor of degree greater than 0. The resultant of two polynomials is zero if and only if they have a nontrivial common factor. Thus, we're specifically looking for when the resultant becomes zero.

Set Up the Resultant Equation

We are given that this resultant \( \text{res}(f, g_{\alpha}) = \alpha^3 + \alpha^2 + \alpha + 1 \). We need to find \( \alpha \) such that this expression is zero. The gcd of \( f \) and \( g_{ix} \) is not 1 if there is a factor in both, implying the resultant is zero.

Solve \( \alpha^3 + \alpha^2 + \alpha + 1 = 0 \)

Now, solve the polynomial equation \( \alpha^3 + \alpha^2 + \alpha + 1 = 0 \) to find the roots. This cubic equation can be factored as \( (\alpha + 1)(\alpha^2 + 1) = 0 \), giving us the solutions.

Find the Roots

The solutions to \((\alpha + 1)(\alpha^2 + 1) = 0\) are \(\alpha = -1\) from \(\alpha + 1 = 0\) and \(\alpha = i, -i\) from \(\alpha^2 + 1 = 0\). These are the values of \(\alpha\) that satisfy the equation.

Key concepts

Monic Polynomials

Monic polynomials are a special class of polynomials where the leading coefficient, the coefficient of the highest degree term, is 1. These polynomials play a crucial role in algebra, particularly because they simplify many expressions and calculations. For example, consider the monic polynomial \(f(x) = x^2 + 3x + 4\). Here, the leading term \(x^2\) has a coefficient of 1, indicating that it is indeed a monic polynomial.

Monic polynomials are particularly useful because they often lead to simpler expressions when finding solutions to problems involving algebraic structures. They ensure that when performing operations such as division, the results are easier to process without extra scaling factors. Furthermore, in computational algebra, using monic polynomials can enhance the efficiency of algorithms by avoiding unnecessary multiplications.

Greatest Common Divisor (GCD)

The Greatest Common Divisor (GCD) of two polynomials is the highest degree polynomial that divides each of the given polynomials without a remainder. This concept is analogous to the GCD of numbers but extends to polynomial expressions.

To compute the GCD of two polynomials, one can use methods similar to those for numbers, such as the Euclidean algorithm. In the context of the exercise, we need to find when the GCD of two given monic polynomials is not 1. This is a sign that the two polynomials share a common factor besides 1, and the GCD signifies the degree of this common factor.

Recognizing when the GCD is not 1 involves examining whether the polynomials share a root or a polynomial factor. This means solving or factoring the polynomial expressions to determine the divisors beyond 1, which influences their dynamics and relationships in algebraic structures.

Resultant of Polynomials

The resultant of two polynomials is a scalar quantity that provides information about the existence of common roots. If two polynomials share a common root, the determinant derived from their resultant is zero. This determinant, formed using a matrix of polynomial coefficients, is a powerful tool in eliminating common factors.

In the given problem, the resultant is expressed as \( \alpha^3 + \alpha^2 + \alpha + 1 \). Finding when this resultant is zero gives us critical information about the shared factors of the polynomials involved. The resultant being zero means that there is a nontrivial common factor between the two polynomials, which is essential for determining when the GCD is not 1.

Thus, evaluating the resultant aids in understanding when the polynomials, despite being initially independent, have intertwined solutions, reflecting on how they intersect in the polynomial space.

Roots of Polynomial Equations

Roots of polynomial equations represent the values of the variable for which the polynomial expression holds a true equal to zero. Solving polynomial equations involves finding these roots, which indicate where the polynomial graph crosses the x-axis.

In the context of the example problem, solving the equation \(\alpha^3 + \alpha^2 + \alpha + 1 = 0\) helps identify the values of \(\alpha\) that cause the resultant to be zero. The roots were found by factoring the cubic polynomial into \((\alpha + 1)(\alpha^2 + 1)\). This factorization reveals the solutions \(\alpha = -1, i, -i\), indicating the precise points where shared roots result in a nontrivial gcd.

These solutions help us understand the behavior and interaction of polynomials, especially in determining how they can potentially simplify or complicate algebraic processes. Finding these roots is an essential step in solving problems involving polynomial gcds and resultants.