Factorization of Polynomials over a field

In summary, the given polynomial x3 - 23x2 - 97x + 291 is irreducible over Z3[x] and Z7[x], as a calculation shows it has no root in these fields. For Z11[x], the polynomial has no roots among 1, 5, 6, or 10, and therefore is also irreducible.
  • #1
lilcoley23@ho
19
0
I don't understand how to factor a polynomial over Z3 [x], Z7 [x], and Z11 [x]

I need to factor the polynomail x3 - 23x2 - 97x + 291

PLEASE HELP!
 
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  • #2
Note that for a polynomial is of degree 2 and 3, reducibility is equivalent to the existence of roots.

mod 3:

X^3-23X^2-97X+291=X^3+X^2+2X=X(X^2+X+2). A calculation shows X^2+X+2 doesn't have a root in Z/3Z. Done.

mod 7:
X^3+5X^2+2X+4. A calculation shows it has no root in Z/7Z. The polynomial is irreducible.

mod 11:
291=5=1*5=10*6. So if the polynomial has a root, it should be 1, 5, 6, or 10. A calculation shows X^3-X^2+2X+5 has no root. The given poly is irreducible.
 
  • #3
I think I might have phrased this question wrong, but I figured it out.

THANKS
 

FAQ: Factorization of Polynomials over a field

1. What is the definition of factorization of polynomials over a field?

The factorization of polynomials over a field is the process of breaking down a polynomial into simpler polynomials that can be multiplied together to get the original polynomial. It is similar to finding the prime factors of a number.

2. Why is factorization of polynomials over a field important?

Factorization of polynomials over a field is important because it helps in simplifying complex expressions and solving polynomial equations. It also allows us to find the roots of a polynomial, which can have important applications in various fields of mathematics and science.

3. What is the difference between factoring over a field and factoring over the integers?

The main difference between factoring over a field and factoring over the integers is the type of numbers used. In factoring over a field, we use numbers from a specific mathematical structure called a field, while in factoring over integers, we use only whole numbers.

4. What are the methods used for factorization of polynomials over a field?

There are several methods for factorization of polynomials over a field, including the quadratic formula, grouping, and factoring by grouping. Other methods such as the rational root theorem and the method of undetermined coefficients can also be used in specific cases.

5. Can all polynomials be factored over a field?

No, not all polynomials can be factored over a field. Some polynomials, known as irreducible polynomials, cannot be broken down into simpler polynomials. These polynomials have no factors other than themselves and the constant term. However, most polynomials can be factored over a field using the appropriate methods.

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