Question

Show (prove) that the set of polynomials of degree less than or
equal to 7 with

real coefficients should be uncountable.

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Answer #1

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).

Show (prove) that the set of sequences of 0s and 1s with only
finitely many
nonzero terms should be countable.
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Prove that the singleton set {0} is a vector subspace of the space
P4(R) of all polynomials of degree at most 3 with real
coefficients.

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 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

Let ℙn be the set of real polynomials of degree at most n, and
write p′ for the derivative of p. Show that
S={p∈ℙ9:p(2)=−1p′(2)}
is a subspace of ℙ9.

8. List all irreducible polynomials with binary coefficients of
degree 4 or less. (Hint: produce a times table that shows the
minimum number of products needed.) Show these as binary numbers
(omitting the indeterminant) and as decimal numbers (interpreting
the binary number into decimal). Is 23 a prime polynomial in this
field?
9. Interpreting these decimal numbers into coefficients of
polynomials with binary coefficients, what is the product of 11 and
10 modulo 31 in GF(2^4) over P = 31?...

Show (prove), from the original definition of the integers, that
subtraction of
integers is well defined.
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Show that the window size must be less than or equal
to half the size of the sequence number space for SR protocols.
Mathematically prove it.
Note: You can do hand write it on a paper and then
upload (submit)

Show that the window size must be less than or equal to half the
size of the sequence number space for SR protocols.
Mathematically prove it.

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