Let P(R#) represent the set of all polynomial functions, functions that can be written in the form anxn + an-1xn-1 + ... + a1x + a0, for some integer n ≥ 0 and with an, an-1, ... a1 and a0 being real numbers, let the operation + represent polynomial addition, and let the operation * represent polynomial multiplication. a. Demonstrate or explain why the system (P(R#), +, *) is a ring, that is, demonstrate or explain why: i. (P(R#), +) is commutative group ii. (P(R#), *) is semi-group iii. The operation * distributes over the operation +.
Let P(R#) represent the set of all polynomial functions, functions that can be written
in the form anxn + an-1xn-1 + ... + a1x + a0, for some integer n ≥ 0 and with an, an-1, ... a1 and
a0 being real numbers, let the operation + represent polynomial addition, and let the
operation * represent polynomial multiplication.
a. Demonstrate or explain why the system (P(R#), +, *) is a ring, that is, demonstrate
or explain why:
i. (P(R#), +) is commutative group
ii. (P(R#), *) is semi-group
iii. The operation * distributes over the operation +.
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Demonstrate or explain why the system (P(R#), +, *) is a ring with identity and is a
commutative ring, that is, demonstrate or explain why:
i. The ring (P(R#), +, *) has an identity element corresponding to the * operation
ii. The ring (P(R#), +, *) is commutative corresponding to the * operation.
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