Computation Question in the Ring of Polynomial with Integer Coefficients

In summary, the conversation discusses the problem of proving that there is no integer m for which 3x2+4x + m is a factor of 6x4+50 in Z[x], where Z[x] is not a field and the division algorithm for polynomials does not guarantee a quotient and remainder. The individual asking the question is seeking guidance on how to write a proof for this problem and is considering working in Q[x] to find a solution.
  • #1
jmjlt88
96
0
I have a quick question. The problem reads:

Prove that there is no integer m such that 3x2+4x + m is a factor of 6x4+50 in Z[x].

Now, Z[x] is not a field. So, the division algorithm for polynomials does not guarantee us a quotient and remainder. When I tried dividing 6x4+50 by 3x2+4x + m, it immediately would have push me out of the integers. So, 3x2+4x + m cannot be a factor no matter what m is.

My question, how can write that nicely in a proof? This may be a silly question. I have finished my first run through of Pinter's A Book of Abstract Algebra and I am now going back, re-writing proofs more concisely, fixing mistakes, and trying ones I skipped or did not "get." I only wrote an explanation similar to the one above for my answer with an attempt at divinding the polynomials.
 
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  • #2
Based on your description, it sounds like you want to work in Q[x]. What does working in Q[x] tell you, and can you relate that fact to the thing you're trying to show?
 

Related to Computation Question in the Ring of Polynomial with Integer Coefficients

1. What is the Ring of Polynomial with Integer Coefficients?

The Ring of Polynomial with Integer Coefficients is a mathematical concept that involves a set of polynomials with integer coefficients. This set forms a ring, which means that it follows certain rules of addition and multiplication, such as closure and associativity. In this ring, the coefficients of the polynomials can only be integers, and the variables can be any mathematical expression.

2. How is computation done in the Ring of Polynomial with Integer Coefficients?

Computation in this ring is done by applying the rules of addition and multiplication to the polynomials. Addition involves adding the coefficients of like terms, while multiplication involves multiplying the coefficients and adding the exponents of the variables. It is important to note that in this ring, the coefficients must always be integers, and any non-integer coefficients must be converted to their nearest integer value.

3. Can polynomials with variables other than integers be included in the Ring of Polynomial with Integer Coefficients?

No, the Ring of Polynomial with Integer Coefficients only includes polynomials with integer coefficients. This restriction is what allows for the ring to follow the rules of addition and multiplication, as non-integer coefficients can lead to inconsistencies and exceptions in these rules.

4. What are some real-world applications of the Ring of Polynomial with Integer Coefficients?

The Ring of Polynomial with Integer Coefficients has various applications in fields such as computer science, engineering, and cryptography. In computer science, it is used in the analysis and optimization of algorithms. In engineering, it is used in signal processing and control systems. In cryptography, it is used in the creation of secure encryption algorithms.

5. What are the advantages of using the Ring of Polynomial with Integer Coefficients?

One advantage of this ring is that it allows for efficient computation of polynomials with integer coefficients, which are commonly encountered in many mathematical problems. Additionally, the ring has well-defined rules and properties, making it a useful tool for solving complex mathematical equations and systems. Its applications in various fields also make it a versatile and valuable concept to understand.

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