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Give an example of each of the following or explain why no such example exists. (a) Two polynomials of degree 1 in Z4x whose product is a polynomial of degree 1. (b) Two polynomials of degree 3 in R[x] whose sum is a polynomial of degree 1. (c) Two polynomials of degree 2 in Z6x whose product is a polynomial of degree 5.

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Answer:

Step-by-step explanation:

(a)

let, f(x) = 2x+1 & g(x) = 2x+2

So, f(x).g(x) = (2x+1).(2x+2) = 4x² + 6x + 2 = 2x+2 in [tex]Z_4[x][/tex]

So, product of two 1 degree polynomials is a 1-degree polynomial in [tex]Z_4[x][/tex]

(b)

let, f(x) = 2x³ - 3x² + 5x + 7 & g(x) = -2x³ + 3x² + x + 3 be two polynomials in R[x]

So, f(x) + g(x) = 6x + 10, which is a polynomial of degree 1 in R[x]

(c)

this is not possible, since, if f & g are 2 polynomials in any polynomial ring S, then, we always have,

deg(f.g) <= deg(f) + deg(g)

So, deg(f.g) = 5 (as desired) but, deg(f) + deg(g) = 2+2 = 4 but, 4 < 5

So, this is not possible.

So, no such example exists.