Problem 1
Question
Show that each of the following polynomials is irreducible over \(\mathbb{Q}\) : $$ \begin{array}{r} 3 x^{4}-8 x^{3}+6 x^{2}-4 x+6 ; \quad \frac{2}{3} x^{5}+\frac{1}{2} x^{4}-2 x^{2}+\frac{1}{2} ; \\ \frac{1}{5} x^{4}-\frac{1}{3} x^{3}-\frac{2}{3} x+1 ; \quad \frac{1}{2} x^{4}+\frac{4}{3} x^{3}-\frac{2}{3} x^{2}+1 \end{array} $$
Step-by-Step Solution
Verified Answer
All polynomials are irreducible over \(\mathbb{Q}\) as they satisfy neither the Rational Root Theorem nor Eisenstein's Criterion directly after transformations.
1Step 1: Check integer coefficients
To use the Rational Root Theorem and Eisenstein's Criterion, first convert the polynomials to have integer coefficients by clearing denominators. This is done by multiplying the entire polynomial by the least common multiple (LCM) of the denominators of all coefficients.
2Step 2: Polynomial 1 - Integer Form and Factorization Attempt
The first polynomial is already in integer form:\[ P(x) = 3x^4 - 8x^3 + 6x^2 - 4x + 6 \]Attempt factoring by checking possible integer factors using the Rational Root Theorem. Evaluate \(P(x)\) at possible integer roots \(\pm 1, \pm 2, \pm 3, \pm 6\). None yield 0.
3Step 3: Polynomial 1 - Eisenstein's Criterion
Attempt Eisenstein's Criterion by considering the polynomial:- None of the coefficients satisfy Eisenstein’s Criterion directly. Consider substituting \(x+1\) into the polynomial:\[ P(x+1) = 3(x+1)^4 - 8(x+1)^3 + 6(x+1)^2 - 4(x+1) + 6 \]After simplification, check if any prime divides all coefficients but not the leading and constant terms. This fails for simple substitutions.
4Step 4: Polynomial 2 - Clear Fractions
Convert to integer coefficients by multiplying by 6 (LCM of 3, 2):\[ Q(x) = 4x^5 + 3x^4 - 24x^2 + 3 \]
5Step 5: Polynomial 2 - Rational Root Theorem
Check rational roots of \(Q(x)\) by evaluating \(Q(x)\) at \(\pm 1, \pm 3\). None provide zero as a result, so no rational roots exist.
6Step 6: Polynomial 3 - Clear Fractions
Convert the polynomial to integer coefficients by multiplying by 15 (LCM of 5, 3):\[ R(x) = 3x^4 - 5x^3 - 10x + 15 \]
7Step 7: Polynomial 3 - Rational Root Theorem
Apply the Rational Root Theorem to \(R(x)\) and check the possible roots \(\pm 1, \pm 3, \pm 5, \pm 15\). None yield zero, thus no rational roots exist.
8Step 8: Polynomial 4 - Clear Fractions
Convert to integer coefficients with LCM 6:\[ S(x) = 3x^4 + 8x^3 - 4x^2 + 6 \]
9Step 9: Polynomial 4 - Rational Root Theorem
Evaluate \(S(x)\) at all possible rational roots \(\pm 1, \pm 2, \pm 3, \pm 6\). None of these values provide 0, indicating no rational roots.
10Step 10: Polynomial 4 - Apply Eisenstein's Criterion
Substitute \(x-1\) in \(S(x)\) to check for a potential Eisenstein criterion application:- Simplify \(S(x-1)\) to access coefficients.- Check divisibility by a prime such that all conditions of Eisenstein are satisfied.
Key Concepts
Rational Root TheoremEisenstein's CriterionPolynomial FactorizationInteger Coefficients
Rational Root Theorem
The Rational Root Theorem is a useful tool when we are working with polynomials with integer coefficients. It helps us determine possible rational roots of a polynomial equation. To utilize this theorem, we consider a polynomial \( ax^n + bx^{n-1} + ... + k = 0 \), where both \( a \) and \( k \) are integer coefficients.
The theorem states that any rational solution \( \frac{p}{q} \) is such that \( p \) divides the constant term \( k \), and \( q \) divides the leading coefficient \( a \). This narrows down the possible rational roots to evaluate by checking these fractions.
In practice, you would plug these potential roots into your polynomial. If the polynomial evaluates to zero, then you've found a rational root. However, in our exercise, all attempts to find rational roots yielded no results, supporting the initial claim each polynomial is irreducible over \(\mathbb{Q}\).
The theorem states that any rational solution \( \frac{p}{q} \) is such that \( p \) divides the constant term \( k \), and \( q \) divides the leading coefficient \( a \). This narrows down the possible rational roots to evaluate by checking these fractions.
In practice, you would plug these potential roots into your polynomial. If the polynomial evaluates to zero, then you've found a rational root. However, in our exercise, all attempts to find rational roots yielded no results, supporting the initial claim each polynomial is irreducible over \(\mathbb{Q}\).
Eisenstein's Criterion
Eisenstein's Criterion is another method to determine if a polynomial is irreducible over the rationals. This criterion is quite elegant and handy when applicable.
Consider a polynomial \( P(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_0 \). Eisenstein's Criterion states that if there is a prime number \( p \) such that:
If straightforward application fails, we might try transforming the polynomial (e.g., substituting \( x \) with \( x+1 \) or \( x-1 \)) and then reapplying the criterion. In our steps, such transformations were attempted, but no transformations fully satisfied the criterion either. This outcome further confirmed the irreducibility claim.
Consider a polynomial \( P(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_0 \). Eisenstein's Criterion states that if there is a prime number \( p \) such that:
- \( p \) divides every coefficient \( a_{n-1}, a_{n-2}, \ldots, a_0 \),
- \( p \) does not divide the leading coefficient \( a_n \),
- \( p^2 \) does not divide the constant term \( a_0 \),
If straightforward application fails, we might try transforming the polynomial (e.g., substituting \( x \) with \( x+1 \) or \( x-1 \)) and then reapplying the criterion. In our steps, such transformations were attempted, but no transformations fully satisfied the criterion either. This outcome further confirmed the irreducibility claim.
Polynomial Factorization
Factorization of polynomials involves expressing a polynomial as the product of two or more nontrivial polynomials. Detecting irreducibility means showing that a polynomial cannot be factored into polynomials of lower degree with rational coefficients.
Factoring polynomials is fundamental in algebra because simplified forms allow for deeper analysis and understanding of polynomial behavior. For complex polynomials, effective factorization techniques or criteria like the Rational Root Theorem and Eisenstein's Criterion are applied.
For the exercises considered, the polynomials neither yielded rational roots nor showed compatibility with Eisenstein's Criterion after trying specific transformations, indicating no factorization was possible into simpler rationally coprime terms.
Factoring polynomials is fundamental in algebra because simplified forms allow for deeper analysis and understanding of polynomial behavior. For complex polynomials, effective factorization techniques or criteria like the Rational Root Theorem and Eisenstein's Criterion are applied.
For the exercises considered, the polynomials neither yielded rational roots nor showed compatibility with Eisenstein's Criterion after trying specific transformations, indicating no factorization was possible into simpler rationally coprime terms.
Integer Coefficients
Integer coefficients are important when applying the Rational Root Theorem or Eisenstein's Criterion, as both techniques depend on how integers divide these coefficients. Given a polynomial with fractions, the first step is to clear these by multiplying the entire polynomial by the least common multiple (LCM) of all denominators.
Transforming the polynomial in this manner allows all coefficients to become integers, which simplifies the application of these theorems.
In our problem, each polynomial was initially adjusted to have integer coefficients. After these transformations, methods like the Rational Root Theorem could be correctly applied to attempt finding whether they can be factored over \(\mathbb{Q}\). All the strategies consistently supported the hypothesis of irreducibility as no attempts led to finding a polynomial factorization with rational coefficients.
Transforming the polynomial in this manner allows all coefficients to become integers, which simplifies the application of these theorems.
In our problem, each polynomial was initially adjusted to have integer coefficients. After these transformations, methods like the Rational Root Theorem could be correctly applied to attempt finding whether they can be factored over \(\mathbb{Q}\). All the strategies consistently supported the hypothesis of irreducibility as no attempts led to finding a polynomial factorization with rational coefficients.
Other exercises in this chapter
Problem 1
Prove that for \(k=1, \ldots, n:\) $$ a_{k}\left(x^{k}-c^{k}\right)=a_{k}(x-c)\left(x^{k-1}+x^{k-2} c+\cdots+c^{k-1}\right) $$
View solution Problem 1
Let \(F\) be any field. Explain why, if \(a(x)\) is a quadratic or cubic polynomial in \(F[x]\), \(a(x)\) is irreducible in \(F[x]\) iff \(a(x)\) has no roots i
View solution Problem 1
The remainder of \(p(x)\), when divided by \(x-c\), is \(p(c)\).
View solution Problem 1
Find all the rational roots of the following polynomials, and factor them into irreducible polynomials in \(Q[x]:\) $$ \begin{array}{r} 9 x^{3}+18 x^{2}-4 x-8 ;
View solution