Therefore, the solutions are x = -1, 1, and 2 (mod 5).

In summary, there are exactly three solutions for the equation 2x^3+x^2+3x-1 = 0 (mod 5). These solutions are x = 2, 1, and 4.
  • #1
Suvadip
74
0
Show that 2x^3+x^2+3x-1 = 0 (mod 5)
has exactly three solutionsHow to proceed with it?
 
Last edited:
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  • #2
Substitute $x=0,1,2,3,4$.
 
  • #3
Hello, suvadip!

I suppose someone wants to see all the steps.
[tex]\text{Show that }\,2x^3+x^2+3x-1\:\equiv\:0\text{ (mod 5)}[/tex]
[tex]\text{has exactly three solutions.}[/tex]

We have: .[tex]2x^3+x^2 + 3x-1\:\equiv\:0\text{ (mod 5)}[/tex]

Then: .[tex]2x^3 + x^2 - 2x - 1 \:\equiv\:0\text{ (mod 5)}[/tex]

Factor: .[tex]x^2(2x+1) - (2x+1) \:\equiv\;0\text{ (mod 5)}[/tex]

Factor: .[tex](2x+1)(x^2-1) \:\equiv\:0\text{ (mod 5)}[/tex]

Factor: .[tex](2x+1)(x-1)(x+1) \:\equiv\:0\text{ (mod 5)}[/tex][tex]2x+1\:\equiv\:0 \text{ (mod 5)}\quad\Rightarrow\quad 2x \:\equiv\:-1 \text{ (mod 5)}[/tex]

. . [tex]2x \:\equiv\:4 \text{ (mod 5)} \quad\Rightarrow\quad \boxed{x \:\equiv\:2\text{ (mod 5)}} [/tex][tex]x-1\:\equiv\:0\text{ (mod 5)} \quad\Rightarrow\quad \boxed{x \:\equiv\:1\text{ (mod 5)}}[/tex][tex]x+1\:\equiv\:0\text{ (mod 5)} \quad\Rightarrow\quad x \:\equiv\:-1\text{ (mod 5)}[/tex]

. . [tex]\boxed{x\:\equiv\:4\text{ (mod 5)}}[/tex]
 

FAQ: Therefore, the solutions are x = -1, 1, and 2 (mod 5).

What is number theory?

Number theory is a branch of mathematics that deals with the properties and relationships of integers. It focuses on studying patterns and structures within the set of whole numbers and their properties, such as prime numbers, divisibility, and modular arithmetic.

How is number theory used to solve problems?

Number theory can be used to solve a variety of problems, including cryptography, coding theory, and optimization problems. It provides tools and techniques for analyzing and manipulating numbers, which can be applied to real-world problems in various fields.

What is a solution using number theory?

A solution using number theory refers to the application of number theory concepts and techniques to solve a particular problem or puzzle. It may involve using mathematical properties and relationships of numbers to find patterns or solutions.

Can number theory be used in computer science?

Yes, number theory has many applications in computer science, particularly in the fields of cryptography and coding theory. It provides the foundation for algorithms and protocols that secure data transmission and storage.

Is number theory still relevant in modern mathematics?

Yes, number theory remains a relevant and active area of research in modern mathematics. It has applications in various fields, such as computer science, physics, and engineering, and continues to contribute to our understanding of number systems and their properties.

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