Problem 5
Question
Prove the following: In \(\mathbb{Z}_{3}[x], x+2\) is a factor of \(x^{m}+2\), for all \(m .\) In \(\mathbb{Z}_{n}[x]\), \(x+(n-1)\) is a factor of \(x^{m}+(n-1)\), for all \(m\) and \(n\).
Step-by-Step Solution
Verified Answer
In \( \mathbb{Z}_3[x] \), \( x+2 \) divides \( x^m+2 \) for all \( m \); this generalizes to \( x+(n-1) \) being a factor of \( x^m+(n-1) \) in \( \mathbb{Z}_n[x] \) for all \( m \), \( n \).
1Step 1: Use Remainder Theorem
According to the Remainder Theorem, a polynomial \( f(x) \) has a factor \( x - k \) if and only if \( f(k) = 0 \). We need to check if \( x+2 \) is a factor of \( x^m + 2 \) in \( \mathbb{Z}_3[x] \). The equivalent form of \( x + 2 \) in \( \mathbb{Z}_3[x] \) is \( x - 1 \) because \( 2 \equiv -1 \pmod{3} \).
2Step 2: Evaluate the Polynomial for Specific Case
We substitute \( x = 1 \) into the polynomial because \( x - 1 \equiv x + 2 \pmod{3} \). We compute \( f(1) = 1^m + 2 \equiv 1 + 2 \equiv 3 \equiv 0 \pmod{3} \).
3Step 3: Conclude First Part
Since \( f(1) = 0 \) in \( \mathbb{Z}_3 \), \( x^m + 2 \) is divisible by \( x+2 \) in \( \mathbb{Z}_3[x] \) for any \( m \).
4Step 4: Generalize to Any \( \mathbb{Z}_n \)
For the polynomial \( x^m + (n-1) \) in \( \mathbb{Z}_n[x] \), consider the potential factor \( x + (n-1) \equiv x-(1) \). Evaluate \( f(1) \equiv 1^m + (n-1) \equiv 1 + (n-1) \equiv n \equiv 0 \pmod{n} \).
5Step 5: Conclude General Case
Since \( f(1) \equiv 0 \pmod{n} \), \( x^m + (n-1) \) is divisible by \( x + (n-1) \) in \( \mathbb{Z}_n[x] \) for any \( m \) and \( n \).
Key Concepts
Remainder TheoremPolynomialsModulo ArithmeticFactors
Remainder Theorem
The Remainder Theorem is a helpful tool in algebra that connects polynomial division with evaluation. It states that for a given polynomial \( f(x) \), the remainder of \( f(x) \) divided by \( x - k \) is \( f(k) \). This means if you substitute \( k \) into the polynomial, the result is the remainder.
For practical applications, the Remainder Theorem is used to check if \( x - k \) is a factor of \( f(x) \). Specifically, \( x - k \) is a factor if the remainder is zero, or put another way, if \( f(k) = 0 \). In our exercise, we use this theorem to prove that certain polynomial expressions are divisible by specific factors. This simplifies solving polynomials by avoiding long division.
For practical applications, the Remainder Theorem is used to check if \( x - k \) is a factor of \( f(x) \). Specifically, \( x - k \) is a factor if the remainder is zero, or put another way, if \( f(k) = 0 \). In our exercise, we use this theorem to prove that certain polynomial expressions are divisible by specific factors. This simplifies solving polynomials by avoiding long division.
- Use substitution to find remainders quickly.
- Verify polynomial factors by checking if \( f(k) = 0 \).
Polynomials
Polynomials are expressions made from variables and coefficients, using only addition, subtraction, multiplication, and non-negative integer exponents. They can look like \( x^2 + 3x + 2 \), for example.
In algebra, polynomials are key structures because they represent a broad class of functions. The degree of a polynomial is the highest power of the variable that appears.
Understanding polynomials also helps with factorization and simplification tasks, pivotal skills in algebraic problems.
In algebra, polynomials are key structures because they represent a broad class of functions. The degree of a polynomial is the highest power of the variable that appears.
- The term with the highest degree determines the polynomial’s behavior at large values of variables.
- Polynomials are versatile: they can model natural processes, fit curves, and solve equations.
Understanding polynomials also helps with factorization and simplification tasks, pivotal skills in algebraic problems.
Modulo Arithmetic
Modulo arithmetic, often called clock arithmetic, involves numbers wrapping around after they reach a certain value, known as the modulus. For instance, in \( \mathbb{Z}_3 \), numbers are considered equivalent if they have the same remainder when divided by 3.
Given its circular nature, it is especially useful in solving problems involving cycles and repetitions. In our exercise, we work in \( \mathbb{Z}_n \), where each number wraps around at \( n \).
By using modulo arithmetic, we demonstrate properties of polynomials that hold across periodic structures.
Given its circular nature, it is especially useful in solving problems involving cycles and repetitions. In our exercise, we work in \( \mathbb{Z}_n \), where each number wraps around at \( n \).
- It simplifies calculations by considering only the remainder.
- It's used to solve congruences and find patterns in number sequences.
By using modulo arithmetic, we demonstrate properties of polynomials that hold across periodic structures.
Factors
In algebra, factors are numbers or expressions that multiply together to give another number or expression. Finding factors helps in simplifying and solving equations.
For a polynomial, a factor is a polynomial of lower degree that divides it exactly, leaving no remainder. In our problem, we confirm whether expressions like \( x+2 \) divide \( x^m + 2 \) with no remainder.
Mastering factor finding is crucial for advanced algebraic manipulation, allowing transformations and solutions of complex problems.
For a polynomial, a factor is a polynomial of lower degree that divides it exactly, leaving no remainder. In our problem, we confirm whether expressions like \( x+2 \) divide \( x^m + 2 \) with no remainder.
- Finding factors helps in polynomial division and simplification.
- It's used in solving algebraic equations and identities.
Mastering factor finding is crucial for advanced algebraic manipulation, allowing transformations and solutions of complex problems.
Other exercises in this chapter
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