/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 4 Let \(f(x)=\left(2 x^{2}+1\right... [FREE SOLUTION] | 91影视

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Chapter 17: Problem 4

Let \(f(x)=\left(2 x^{2}+1\right)\left(5 x^{3}-5 x+3\right)(4 x-3) \in \mathbf{Z}_{7}[x]\). Write \(f(x)\) as the product of a unit and three monic polynomials.

Short Answer

Expert verified
The given function \(f(x)\) can be written as the product of a unit and three monic polynomials as \(f(x)=4(x^2+4)(x^3+x^2+x+6)(x-1)\).

Step by step solution

01

Convert to monic polynomial

First convert the given polynomial \(f(x)=(2 x^{2}+1)(5 x^{3}-5 x+3)(4 x-3)\) into a monic polynomial by dividing it by the leading coefficient of each term. The factorization of \(f(x)\) can be rewritten as \(f(x)=(2x^2+1)(x^3+2x^2+x+6)(4x-3)\). The reduction modulo 7 allows a simplification of the second equation, as \(-5x = 2x (mod 7)\) and \(-5 = 2 (mod 7)\). Similarly, \(4x = -3x (mod 7)\) allows a simplification of the third equation.
02

Rewrite the polynomial

Each term of the polynomial can then be rewritten to reflect the changes made in the first step. This results in the equation \(f(x)=(2x^2+1)(x^3+2x^2+x+6)(-3x+3)\).
03

Factorize the polynomial

The last step is then to divide each term by its leading coefficient, this will result in a unit and three monic polynomials. The unit factor here will be \(2*(-3)*(-3) = -18 = 4 (mod 7)\). The three monic polynomials are \(x^2+4\), \(x^3+x^2+x+6\), and \(x-1\).
04

Form the final expression

Finally we form the expression for \(f(x)\) as the product of the unit and three monic polynomials. This will be \(f(x)=4(x^2+4)(x^3+x^2+x+6)(x-1)\).

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Most popular questions from this chapter

a) List the points and lines in \(A P\left(\mathbf{Z}_{3}\right)\). How many parallel classes are there for this finite geometry? What are the parameters for the associated balanced incomplete block design? b) List the points and lines for the projective plane that arises from \(A P\left(Z_{3}\right)\). Determine

Determine all polynomials \(f(x) \in \mathbf{Z}_{2}[x]\) such that \(1 \leq\) degree \(f(x) \leq 3\) and \(f(x)\) is irreducible (over \(\mathbf{Z}_{2}\) ).

\text { Construct a finite field of } 25 \text { elements. }

A \((v, b, r, k, \lambda)\)-design is called a triple system if \(k=3\). When \(k=3\) and \(\lambda=1\) we call the design a Steiner triple system. a) Prove that in every triple system, \(\lambda(v-1)\) is even and \(\lambda v(v-1)\) is divisible by \(6 .\) b) Prove that in every Steiner triple system, \(v\) is congruent to 1 or 3 modulo \(6 .\)

\text { How many monic polynomials in } \mathbf{Z}_{7}[x] \text { have degree } 5 ?
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