Let f (x) = 5x2 + 2x + 1, g(x) = 5x2 + 7x + 3 b e elements of the p olynomialring Z9[x]. Find f (x) + g(x) and f (x)g(x) in Z9[x].

To find f(x) + g(x) and f(x)g(x) in Z9[x], we first need to define the functions f(x) and g(x).

f(x) = 5x^2 + 2x + 1 g(x) = 5x^2 + 7x + 3

To find f(x) + g(x), we simply add the corresponding terms of f(x) and g(x):

f(x) + g(x) = (5x^2 + 5x^2) + (2x + 7x) + (1 + 3) = 10x^2 + 9x + 4

However, since we are in Z9[x], we need to reduce each coefficient modulo 9:

10x^2 + 9x + 4 = 1x^2 + 0x + 4 = x^2 + 4

So, f(x) + g(x) = x^2 + 4 in Z9[x].

To find f(x)g(x), we multiply the polynomials f(x) and g(x):

f(x)g(x) = (5x^2 + 2x + 1)(5x^2 + 7x + 3) = 25x^4 + 35x^3 + 15x^2 + 10x^2 + 14x + 3x + 5x^2 + 7x + 3 = 25x^4 + 35x^3 + 30x^2 + 17x + 3

Again, we need to reduce each coefficient modulo 9:

25x^4 + 35x^3 + 30x^2 + 17x + 3 = 7x^4 + 8x^3 + 3x^2 + 8x + 3

So, f(x)g(x) = 7x^4 + 8x^3 + 3x^2 + 8x + 3 in Z9[x].