Compute the sum and product of p(x) and q(x) (a) p(x)=1+x+2...

Compute the sum and product of $p(x)$ and $q(x)$
(a) $p(x)=1+x+2 x^{2}, q(x)=3 x^{2}+x^{3},$ where both polynomials are in $\mathbb{Z}_{5}[x]$.
(b) $p(x)=1+4 x^{2}+3 x^{3}+2 x^{4}, q(x)=5+2 x^{2}+x^{3},$ where both polynomials are in $\mathbb{Z}_{6}[x]$.

Key Concepts

Modular Arithmetic

Modular arithmetic is the system of arithmetic for integers where numbers wrap around after reaching a certain value, known as the modulus. In this framework, operations such as addition, subtraction, and multiplication are performed and then reduced modulo n, ensuring that all results remain within a set of fixed equivalence classes. This concept is critical when performing calculations in rings like Z5 or Z6, as it governs how the coefficients of polynomials are managed.

Polynomial Arithmetic

Polynomial arithmetic involves carrying out operations like addition, subtraction, and multiplication on polynomials. When these polynomials have coefficients in a modular system, each arithmetic operation on the coefficients is followed by a reduction modulo n. This ensures that the result abides by the rules of the modular system, making it a central idea for computing the sum and product of polynomials in rings such as Z5[x] or Z6[x].

Polynomial Rings over Finite Rings

A polynomial ring over a finite ring consists of polynomials whose coefficients come from a finite ring, such as Z5 or Z6. In these rings, the usual operations of polynomial addition and multiplication are performed with an additional step of reducing coefficients modulo n. Understanding the structure of polynomial rings is crucial, because the properties of the underlying ring (for example, whether it is a field or contains zero divisors) directly influence the behavior and factorization properties of the polynomials within the ring.

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