Question

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).

Answer #1

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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.
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 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/ 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 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: 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.
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
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 ∈ 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 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.

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