Solving for the 0's of a polynomial in a ring

In summary, a polynomial is an algebraic expression made up of terms containing constants, variables, and exponents. Solving for the 0's of a polynomial means finding the values of the variable that make the polynomial equal to 0, also known as its roots. A ring in mathematics is a set of elements closed under addition and multiplication, used to study polynomials and other structures. To solve for the 0's of a polynomial in a ring, various methods such as factoring and the quadratic formula can be used. Solving for the 0's of a polynomial is important as it provides information about its behavior and properties, and is useful in solving equations and systems in mathematics and science.
  • #1
Werg22
1,431
1
Say I have x^2 + 1 = 0 mod 7, it is clear that adding a linear polynomial congruent to 0 will give away the solution; problem is, what polynomial is that? Trial and error is quite tedious, though I doubt it, is there a direct way?
 
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  • #2
I presume you mean that you're operating in the ring of integers mod 7. There are only 7 elements in that field. Finding the zeroes/roots should be trivial.
 
  • #3
I realized that 30 sec after I made that thread :redface:
 
  • #4
some thinking seems in order before speaking, or asking. but with me asking often stimulates thought.
 

What is a polynomial?

A polynomial is an algebraic expression made up of terms that can contain constants, variables, and exponents. It is usually written in the form of ax^n + bx^(n-1) + cx^(n-2) + ... + k, where a, b, c, and k are constants, x is the variable, and n is a non-negative integer.

What does it mean to solve for the 0's of a polynomial?

Solving for the 0's of a polynomial means finding the values of the variable that make the polynomial equal to 0. These values are also known as the roots or solutions of the polynomial.

What is a ring in mathematics?

In mathematics, a ring is a set of elements that is closed under two operations, usually addition and multiplication. These operations follow certain rules, such as commutativity and distributivity. Rings are used in algebra and abstract algebra to study the properties of polynomials and other mathematical structures.

How do you solve for the 0's of a polynomial in a ring?

To solve for the 0's of a polynomial in a ring, you can use various methods such as factoring, the quadratic formula, or the rational roots theorem. These methods involve manipulating the polynomial in a way that makes it easier to find the solutions.

Why is solving for the 0's of a polynomial important?

Solving for the 0's of a polynomial allows us to find the points where the polynomial intersects the x-axis, which can provide valuable information about the behavior and properties of the polynomial. It is also essential for solving equations and systems of equations in various fields of mathematics and science.

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