Problem 3 - A Book of Abstract Algebra

Question

Find the quotient and remainder when \(x^{3}+2\) is divided by \(2 x^{2}+3 x+4\) in \(\mathbb{Z}[x]\) in \(\mathbb{Z}_{3}[x]\), and in \(\mathbb{Z}_{5}[x]\) We call \(b(x)\) a factor of \(a(x)\) if \(a(x)=b(x) q(x)\) for some \(q(x)\), that is, if the remainder when \(a(x)\) is divided by \(b(x)\) is equal to zero.

Step-by-Step Solution

Answer

In \(\mathbb{Z}[x]\), remainder is \(x^3+2\). In \(\mathbb{Z}_3[x]\), quotient \(2x\), remainder \(x+2\). In \(\mathbb{Z}_5[x]\), quotient \(3x\), remainder \(x+3\).

Step 1: Understand the Polynomial Division Context

We have a polynomial division problem where we need to find the quotient and remainder when dividing the polynomial \(x^3 + 2\) by \(2x^2 + 3x + 4\). We are asked to perform this division in three different rings: \(\mathbb{Z}[x]\), \(\mathbb{Z}_3[x]\), and \(\mathbb{Z}_5[x]\). This means we need to do the division normally in the first case, and apply modular arithmetic in the other two.

Step 2: Polynomial Division in \(\mathbb{Z}[x]\)

In \(\mathbb{Z}[x]\), we perform polynomial division as usual. We aim to express \(x^3 + 2\) as \((2x^2 + 3x + 4)q(x) + r(x)\) where the degree of \(r(x)\) is less than the degree of \(2x^2 + 3x + 4\). 1. Divide the leading term \(x^3\) by the leading term \(2x^2\), giving \(\frac{1}{2}x\).2. This fraction is problematic in \(\mathbb{Z}[x]\), indicating no meaningful integer quotient and thus very likely reducing to \(0\) quotient and \(x^3 + 2\) remainder.

Step 3: Polynomial Division in \(\mathbb{Z}_{3}[x]\)

Now conduct the division with coefficients reduced modulo 3. Thus, the polynomials become:- Dividend: \(x^3 + 2 \rightarrow x^3 + 2\)- Divisor: \(2x^2 + 3x + 4 \rightarrow 2x^2 + 0x + 1\) since \(3x \equiv 0x\) and \(4 \equiv 1\) (mod 3).Begin the division:1. Divide \(x^3\) by \(2x^2\), resulting in \(\frac{1}{2}x \equiv 2x\) mod 3 (since inverse of 2 is 2 mod 3).2. Multiply \(2x\) by \(2x^2\), then subtract from \(x^3 + 0x + 2\). This results in a remainder of \(1x + 0\).3. Repeat this subtraction until degree condition is fulfilled, getting a quotient of \(2x\) and a remainder of \(x + 2\).

Step 4: Polynomial Division in \(\mathbb{Z}_{5}[x]\)

In \(\mathbb{Z}_{5}[x]\), we reduce coefficients modulo 5. The polynomials become:- Dividend: \(x^3 + 2 \rightarrow x^3 + 2\)- Divisor: \(2x^2 + 3x + 4 \rightarrow 2x^2 + 3x + 4\).Proceed with division:1. Leading term \(x^3\) divided by \(2x^2\) gives \(\frac{1}{2}x \equiv 3x\) mod 5 (since inverse of 2 is 3 mod 5).2. Subtract \(3x(2x^2 + 3x + 4)\) from \(x^3 + 0x^2 + 0x + 2\) to find the remainder.3. Resulting in a division with quotient \(3x\) and remainder \(x + 3\).

Step 5: Compile the Results

In all three rings, we've completed the polynomial division: - In \(\mathbb{Z}[x]\): Remainder \(x^3 + 2\), no quotient (meaningful integer terms).- In \(\mathbb{Z}_{3}[x]\): Quotient \(2x\), remainder \(x + 2\).- In \(\mathbb{Z}_{5}[x]\): Quotient \(3x\), remainder \(x + 3\).

Key Concepts

Remainder Theorem

The Remainder Theorem is a handy tool when dealing with polynomial division. It states that the remainder of the division of a polynomial \(f(x)\) by a linear divisor \(x - c\) is simply \(f(c)\). This means if you evaluate the polynomial at \(x = c\), you get the remainder of that particular division.

This theorem is particularly useful when dealing with polynomials and can save time for evaluations in specific cases where the divisor is of the form \(x-c\). However, in our exercise, the divisor is not a linear polynomial. Thus, the Remainder Theorem directly does not apply. Instead, we perform normal polynomial division to attain quotient and remainder.

• Apply the theorem for linear divisors only.
• Evaluate polynomial at the root \(c\) to find remainder.

Though not directly used here, understanding the Remainder Theorem gives insights into the behavior of polynomials during division.

Modular Arithmetic

Modular arithmetic involves numbers wrapping around upon reaching a certain value, known as the modulus. In this exercise, we conduct divisions in different modular rings: \(\mathbb{Z}_3[x]\) and \(\mathbb{Z}_5[x]\).

In these modular rings, every coefficient of the polynomial is replaced by its equivalent modulo the given base.

• In \(\mathbb{Z}_3[x]\), 3 wraps to 0, 4 becomes 1, etc.
• In \(\mathbb{Z}_5[x]\), 5 wraps to 0, leaving numbers like 6 to wrap to 1, etc.

This wrapping allows us to handle coefficients within these set limitations, simplifying the division because it alters the calculation of the leading quotient term.
All operations (addition, subtraction, multiplication, and reduction) must respect this wrapping, which alters normal arithmetic operations in modular settings.

Polynomial Rings

Polynomial Rings \(\mathbb{Z}[x]\), \(\mathbb{Z}_3[x]\), and \(\mathbb{Z}_5[x]\) are sets of polynomials with coefficients in the respective integer or modular environments.

These rings facilitate polynomial arithmetic, respecting the rules of the coefficients used:

• In \(\mathbb{Z}[x]\), all integer polynomials are used.
• In \(\mathbb{Z}_3[x]\), only integers 0, 1, 2 are coefficients (mod 3).
• In \(\mathbb{Z}_5[x]\), coefficients are integers 0 to 4 (mod 5).

In the context of division, the choice of polynomial ring impacts the result significantly as it determines how coefficients are treated.
For example, division by a non-zero number always yields another non-zero number unless it wraps around to 0 in modular rings, affecting both the quotient and remainder drastically.

Quotient and Remainder

Polynomial division involves finding two key outputs: the quotient \(q(x)\) and the remainder \(r(x)\). The relationship is expressed as:
\[ f(x) = d(x)q(x) + r(x) \]where \( r(x) \) has a degree lower than \( d(x) \).

In our example, the divisions were conducted in three different rings:

• In \(\mathbb{Z}[x]\): No meaningful quotient, remainder is the dividend itself.
• In \(\mathbb{Z}_3[x]\): Quotient is \(2x\), remainder is \(x + 2\).
• In \(\mathbb{Z}_5[x]\): Quotient is \(3x\), remainder is \(x + 3\).

The key is determining when the degree condition for \(r(x)\) is met, signalling the end of the division process.

This concept highlights the differences between conducting division in standard and modular conditions, influencing both outputs.

Factor Theorem

The Factor Theorem is closely related to the Remainder Theorem and deals with roots and factors of polynomials. It states that a polynomial \(f(x)\) has a root \(x = c\) if and only if \((x - c)\) is a factor of \(f(x)\), meaning \(f(c) = 0\).

While this theorem does not directly apply to the exercise at hand since our divisor is not of the form \((x - c)\), understanding this theorem can be highly useful.

• Recognizes roots: Helps verify if a particular value of \(x\) is indeed a root of the polynomial.
• Simplifies factorization: Once roots are determined, simplifies obtaining linear factors quickly.

This theorem is crucial for learning about polynomial structures and how one can deduce factors and roots from polynomial division effectively.