Question

Question:

1. Find a polynomial of positive degree in Z₉[X].
2. Show that every polynomial (except the constant polynomials 3 and 6) in Z₉[X] can be written as the product of two polynomials of positive degree

Short answer

Answer:

1. It is proved that the required polynomial is 1+3X.
2. It is proved that every polynomial in Z₉[X] can be written as the product of two polynomials of positive degree.

Step-by-step solution

Property of Polynomial:

If F is any field with a non-constant polynomial p(x), then for some polynomials a(x) and b(x) , we have:

• If p(x)/ a(x)b(x) then, p(x)/a(x) or p(x)/b(x) .
• If p(x)= a(x)b(x) then, either a(x) or b(x) is a non-zero constant polynomial.

Polynomial Operations

Polynomial Operations: