Let
a1, a2, ..., an be distinct n (≥ 2) integers. Consider the
polynomial
f(x) =...
Let
a1, a2, ..., an be distinct n (≥ 2) integers. Consider the
polynomial
f(x) = (x−a1)(x−a2)···(x−an)−1 in Q[x]
(1) Prove that if then f(x) = g(x)h(x)
for some g(x), h(x) ∈ Z[x],
g(ai) + h(ai) = 0 for all i = 1, 2, ..., n
(2) Prove that f(x) is irreducible over Q
1.13. Let a1, a2, . . . , ak be integers with gcd(a1, a2, . ....
1.13. Let a1, a2, . . . , ak be integers with gcd(a1, a2, . . .
, ak) = 1, i.e., the largest
positive integer dividing all of a1, . . . , ak is 1. Prove that
the equation
a1u1 + a2u2 + · · · + akuk = 1
has a solution in integers u1, u2, . . . , uk. (Hint. Repeatedly
apply the extended Euclidean
algorithm, Theorem 1.11. You may find it easier to prove...
(4) Prove that, if A1, A2, ..., An are countable sets, then A1 ∪
A2 ∪...
(4) Prove that, if A1, A2, ..., An are countable sets, then A1 ∪
A2 ∪ ... ∪ An is countable. (Hint: Induction.)
(6) Let F be the set of all functions from R to R. Show that |F|
> 2 ℵ0 . (Hint: Find an injective function from P(R) to F.)
(7) Let X = {1, 2, 3, 4}, Y = {5, 6, 7, 8}, T = {∅, {1}, {4},
{1, 4}, {1, 2, 3, 4}}, and S =...
10. Let P(k) be the following statement: ”Let a1, a2, . . . , ak
be...
10. Let P(k) be the following statement: ”Let a1, a2, . . . , ak
be integers and p be a prime. If p|(a1 · a2 · a3 · · · ak), then
p|ai for some i with 1 ≤ i ≤ k.” Prove that P(k) holds for all
positive integers k
1) Suppose a1, a2, a3, ... is a sequence of integers such that
a1 =1/16 and...
1) Suppose a1, a2, a3, ... is a sequence of integers such that
a1 =1/16 and an = 4an−1. Guess a formula for an and prove that your
guess is correct.
2) Show that given 5 integer numbers, you can always find two of
the numbers whose difference will be a multiple of 4.
3) Four cats and five mice form a row. In how many ways can they
form the row if the mice are always together?
Please help...
let p1(x) = x^2-3x-10 ,p2(x)=x^2-5x+1,p3(x)=x^2+2x+3 and
p4(x)=x+5
a- As an alternative ; we can form additional...
let p1(x) = x^2-3x-10 ,p2(x)=x^2-5x+1,p3(x)=x^2+2x+3 and
p4(x)=x+5
a- As an alternative ; we can form additional equation involving
a1,a2,a3,and a4 using differntiation.explain how how to produce a
system of equation with unknown constant a1,a2,a3,a4. Since there
are 4 unknown ,you will need 4 equattion find them
b- Find Basis for the vector space spanned by
p1(X),p2(x),p3(x),p4(x). express any reductant vector in terms of
linear combination of the others .Note you might want to use row in
the changes?