Question

Let F be a field and let a(x), b(x) be polynomials in F[x]. Let S be...

Let F be a field and let a(x), b(x) be polynomials in F[x]. Let S be the set of all linear combinations of a(x) and b(x). Let d(x) be the monic polynomial of smallest degree in S. Prove that d(x) divides a(x).
Math   Advanced-Math   
0 0
Add a commentTranscribed image text
Next >

Homework Answers

Answer #1

for any doubt write in comment box.

0 0
Know the answer?
Your Answer:

Post as a guest

Your Name:

What's your source?

Earn Coins

Coins can be redeemed for fabulous gifts.

Log In

Sign Up

Not the answer you're looking for?
Ask your own homework help question
Similar Questions
5. Let S be the set of all polynomials p(x) of degree ≤ 4 such that...
5. Let S be the set of all polynomials p(x) of degree ≤ 4 such that p(-1)=0. (a) Prove that S is a subspace of the vector space of all polynomials. (b) Find a basis for S. (c) What is the dimension of S? 6. Let ? ⊆ R! be the span of ?1 = (2,1,0,-1), ?2 =(1,2,-6,1), ?3 = (1,0,2,-1) and ? ⊆ R! be the span of ?1 =(1,1,-2,0), ?2 =(3,1,2,-2). Prove that V=W.
Let F be a field, and let f (x), g(x) ∈ F [x] be nonzero polynomials....
Let F be a field, and let f (x), g(x) ∈ F [x] be nonzero polynomials. Then it must be the case that deg(f (x)g(x)) = deg(f (x)) + deg(g(x)).
Let F be an ordered field.  Let S be the subset [a,b) i.e, {x|a<=x<b, x element of...
Let F be an ordered field.  Let S be the subset [a,b) i.e, {x|a<=x<b, x element of F}. Prove that infimum and supremum exist or do not exist.
1 Approximation of functions by polynomials Let the function f(x) be given by the following: f(x)...
1 Approximation of functions by polynomials Let the function f(x) be given by the following: f(x) = 1/ 1 + x^2 Use polyfit to approximate f(x) by polynomials of degree k = 2, 4, and 6. Plot the approximating polynomials and f(x) on the same plot over an appropriate domain. Also, plot the approximation error for each case. Note that you also will need polyval to evaluate the approximating polynomial. Submit your code and both plots. Make sure each of...
Prove that the set V of all polynomials of degree ≤ n including the zero polynomial...
Prove that the set V of all polynomials of degree ≤ n including the zero polynomial is vector space over the field R under usual polynomial addition and scalar multiplication. Further, find the basis for the space of polynomial p(x) of degree ≤ 3. Find a basis for the subspace with p(1) = 0.
Let f ∈ Z[x] be a nonconstant polynomial. Prove that the set S = {p prime:...
Let f ∈ Z[x] be a nonconstant polynomial. Prove that the set S = {p prime: there exist infinitely many positive integers n such that p | f(n)} is infinite.
Let F be a field and f(x), g(x) ? F[x] both be of degree ? n....
Let F be a field and f(x), g(x) ? F[x] both be of degree ? n. Suppose that there are distinct elements c0, c1, c2, · · · , cn ? F such that f(ci) = g(ci) for each i. Prove that f(x) = g(x) in F[x].
Let F be a field (for instance R or C), and let P2(F) be the set...
Let F be a field (for instance R or C), and let P2(F) be the set of polynomials of degree ≤ 2 with coefficients in F, i.e., P2(F) = {a0 + a1x + a2x2 | a0,a1,a2 ∈ F}. Prove that P2(F) is a vector space over F with sum ⊕ and scalar multiplication defined as follows: (a0 + a1x + a2x^2)⊕(b0 + b1x + b2x^2) = (a0 + b0) + (a1 + b1)x + (a2 + b2)x^2 λ (b0 +...
Let F be a field and Aff(F) := {f(x) = ax + b : a, b...
Let F be a field and Aff(F) := {f(x) = ax + b : a, b ∈ F, a ≠ 0} the affine group of F. Prove that Aff(F) is indeed a group under function composition. When is Aff(F) abelian?
Let p be an odd prime. Let f(x) ∈ Q(x) be an irreducible polynomial of degree...
Let p be an odd prime. Let f(x) ∈ Q(x) be an irreducible polynomial of degree p whose Galois group is the dihedral group D_2p of a regular p-gon. Prove that f (x) has either all real roots or precisely one real root.
ADVERTISEMENT
Need Online Homework Help?

Get Answers For Free
Most questions answered within 1 hours.

Ask a Question
ADVERTISEMENT