Question

Let R[x] be the set of all polynomials (in the variable x) with real coefficients. Show...

Let R[x] be the set of all polynomials (in the variable x) with real coefficients. Show that this is a ring under ordinary addition and multiplication of polynomials.

What are the units of R[x] ?

I need a legible, detailed explaination

Homework Answers

Know the answer?
Your Answer:

Post as a guest

Your Name:

What's your source?

Earn Coins

Coins can be redeemed for fabulous gifts.

Not the answer you're looking for?
Ask your own homework help question
Similar Questions
Let P(R) denote the family of all polynomials (in a single variable x) with real coefficients....
Let P(R) denote the family of all polynomials (in a single variable x) with real coefficients. We have seen that with the operations of pointwise addition and multiplication by scalars, P(R) is a vector space over R. Consider the 2 linear maps D, I : P(R) to P(R), where D is differentiation and I is anti-differentiation. In detail, for a polynomial p = a0+a1x1+...+anxn, we have D(p) = a1+2a2x+....+nanxn-1 and I(p) = a0x+(a1/2)x2+...+(an/(n+1))xn+1. a. Show that D composed with I...
In the ring R[x] of polynomials with real coefficients, show that A = {f 2 R[x]...
In the ring R[x] of polynomials with real coefficients, show that A = {f 2 R[x] : f(0) = f(1) = 0} is an ideal.
Let Z[x] be the ring of polynomials with integer coefficients. Find U(Z[x]), the set of all...
Let Z[x] be the ring of polynomials with integer coefficients. Find U(Z[x]), the set of all units of Z[x].
Let Z_2 [x] be the ring of all polynomials with coefficients in Z_2. List the elements...
Let Z_2 [x] be the ring of all polynomials with coefficients in Z_2. List the elements of the field Z_2 [x]/〈x^2+x+1〉, and make an addition and multiplication table for the field. For simplicity, denote the coset f(x)+〈x^2+x+1〉 by (f(x)) ̅.
Let P4 denote the space of polynomials of degree less than 4 with real coefficients. Show...
Let P4 denote the space of polynomials of degree less than 4 with real coefficients. Show that the standard operations of addition of polynomials, and multiplication of polynomials by a scalar, give P4 the structure of a vector space (over the real numbers R). Your answer should include verification of each of the eight vector space axioms (you may assume the two closure axioms hold for this problem).
Let P be the vector space of all polynomials in x with real coefficients. Does P...
Let P be the vector space of all polynomials in x with real coefficients. Does P have a basis? Prove your answer.
Let P2 denote the vector space of polynomials in x with real coefficients having degree at...
Let P2 denote the vector space of polynomials in x with real coefficients having degree at most 2. Let W be a subspace of P2 given by the span of {x2−x+6,−x2+2x−1,x+5}. Show that W is a proper subspace of P2.
Let V be the set of polynomials of the form ax + (a^2)(x^2), for all real...
Let V be the set of polynomials of the form ax + (a^2)(x^2), for all real numbers a. Is V a subspace of P?
Prove the division algorithm for R[x]. Where R[x] is the set of real polynomials.
Prove the division algorithm for R[x]. Where R[x] is the set of real polynomials.
Let V be the set of all triples (r,s,t) of real numbers with the standard vector...
Let V be the set of all triples (r,s,t) of real numbers with the standard vector addition, and with scalar multiplication in V defined by k(r,s,t) = (kr,ks,t). Show that V is not a vector space, by considering an axiom that involves scalar multiplication. If your argument involves showing that a certain axiom does not hold, support your argument by giving an example that involves specific numbers. Your answer must be well-written.