Does the equation ${x}^{2}\equiv x\cdot x\equiv 2x\cdot 4xb\text{mod}7$ show that factorization of polynomials $b\text{mod}7$ is not unique? Why or why not?

Mary Reyes
2021-12-19
Answered

Does the equation ${x}^{2}\equiv x\cdot x\equiv 2x\cdot 4xb\text{mod}7$ show that factorization of polynomials $b\text{mod}7$ is not unique? Why or why not?

You can still ask an expert for help

Cassandra Ramirez

Answered 2021-12-20
Author has **30** answers

Step 1

${x}^{2}\equiv x\cdot x\equiv 2x\cdot 4xb\text{mod}7$

To show: factorization of polynomials$b\text{mod}7$ is not unique.

Other examples can be taken as

${x}^{2}\equiv 3x\cdot 5xb\text{mod}7$

${x}^{2}\equiv 6x\cdot 6xb\text{mod}7$

Step 2

From the above example it is clear that factorization of polynomials mod 7 is not unique as there are other factorization also.

To show: factorization of polynomials

Other examples can be taken as

Step 2

From the above example it is clear that factorization of polynomials mod 7 is not unique as there are other factorization also.

Orlando Paz

Answered 2021-12-21
Author has **42** answers

Step 1

Given that, the equation is${x}^{2}\equiv x\cdot x\equiv 2x\cdot 4x\left(b\text{mod}7\right)$ .

${x}^{2}\equiv x\cdot x\left(b\text{mod}7\right)$

${x}^{2}\equiv 2x\cdot 4x\left(b\text{mod}7\right)$

From above equation, it is observed that the factorization of the polynomials$\left(b\text{mod}7\right)$ is not unique.

Step 2

From above equation, it is observed that the factorization of the polynomials$\left(b\text{mod}7\right)$ is not unique.

For example:

${x}^{2}\equiv 2x\cdot 4x\left(b\text{mod}7\right)$

${x}^{2}\equiv (2\cdot 4)(x\cdot x)\left(b\text{mod}7\right)$

${x}^{2}\equiv (3\cdot 5)(x\cdot x)\left(b\text{mod}7\right)$

From above example, it is observed that the factorization of the polynomials$\left(b\text{mod}7\right)$ is not unique.

Because,${x}^{2}\equiv 1\left(b\text{mod}7\right)$ can have both solutions $x\equiv \pm 1\left(b\text{mod}7\right)$ . That is, $x\equiv 1\left(b\text{mod}7\right)\text{}{\textstyle \phantom{\rule{1em}{0ex}}}\text{and}{\textstyle \phantom{\rule{1em}{0ex}}}\text{}x\equiv -1\left(b\text{mod}7\right)$ .

Given that, the equation is

From above equation, it is observed that the factorization of the polynomials

Step 2

From above equation, it is observed that the factorization of the polynomials

For example:

From above example, it is observed that the factorization of the polynomials

Because,

asked 2021-06-03

Determine whether the following function is a polynomial function. If the function is a polynomial function, state its degree. If it is not, tell why not. Write the polynomial in standard form. Then identify the leading term and the constant term.

$g(x)=3-\frac{{x}^{2}}{4}$

asked 2021-12-07

The prime factorization of 7007 is

$7\cdot {11}^{3}\cdot 13$

${7}^{3}\cdot 11\cdot 13$

${7}^{2}\cdot 11\cdot 13$

7*11*13

7*11*13

asked 2022-02-04

How do you multiply polynomials (a+3)(a-2)?

asked 2022-02-02

What is Multiplication of Monomials by Polynomials?

asked 2022-02-09

How do you multiply $({x}^{2}+8x+2)(7x+6)$ ?

asked 2021-03-23

To prove: The system of congruences $x\equiv 2\left(b\text{mod}16\right)$ and $x\equiv 3\left(b\text{mod}9\right)$ has no solution.

asked 2021-12-11

Factor completely each of the trinomials and indicate any that are not factorable using integers. $20{x}^{2}-11x-3$