Question

Prove: If f(x) = anx^n + an−1x^n−1 + ··· + a1x + a0 has integer coefficients with an ? 0 ? a0 and there are relatively prime integers p, q ∈ Z with f ? p ? = 0, then p | a0 and q | an . [Hint: Clear denominators.]

Answer #1

Prove that if f(x) =
akx^k
+ak−1x^k+1
+ak−2x^k+2+...+a1x+a0
is a polynomial in Q[x] and ak
̸= 0, and f (x) factors as f
(x) = g(x)h(x),
where g(x) and h(x) are
polynomials in Q[x], then deg f = deg
g+ deg h.

problem 2 In the polynomial ring Z[x], let I = {a0 + a1x + ... +
anx^n: ai in Z[x],a0 = 5n}, that is, the set of all polynomials
where the constant coefficient is a multiple of 5. You can assume
that I is an ideal of Z[x]. a. What is the simplest form of an
element in the quotient ring z[x] / I? b. Explicitly give the
elements in Z[x] / I. c. Prove that I is not a...

Prove the following statements:
1- If m and n are relatively prime,
then for any x belongs, Z there are integers a; b such that
x = am + bn
2- For every n belongs N, the number (n^3 + 2) is not divisible
by 4.

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 p(x) = a0 + a1x + a2x2 and q(x) = b0 + b1x + b2x2 be vectors
in P2 with p, q = a0b0 + a1b1 + a2b2. Determine whether the given
second-degree polynomials form an orthonormal set, and if not, then
apply the Gram-Schmidt orthonormalization process to form an
orthonormal set. (If the set is orthonormal, enter ORTHONORMAL in
both answer blanks.) { 3 (x2−1), 3 (x2 + x + 2)}
u1 =
u2 =

Let p(x) = a0 + a1x + a2x2 and q(x) = b0 + b1x + b2x2 be vectors
in P2 with p, q = a0b0 + a1b1 + a2b2. Determine whether the given
second-degree polynomials form an orthonormal set, and if not, then
apply the Gram-Schmidt orthonormalization process to form an
orthonormal set. (If the set is orthonormal, enter ORTHONORMAL in
both answer blanks.)
{ square root 3 (x2−1), square root 3 (x2 + x + 2)}
u1 =
u2...

4) Let F be a finite field. Prove that there exists an integer n
≥ 1, such that n.1F = 0F . Show further that the smallest positive
integer with this property is a prime number.

Problem 2: (i) Let a be an integer. Prove that 2|a if and only
if 2|a3.
(ii) Prove that 3√2 (cube root) is irrational.
Problem 3: Let p and q be prime numbers.
(i) Prove by contradiction that if p+q is prime, then p = 2 or q
= 2
(ii) Prove using the method of subsection 2.2.3 in our book that
if
p+q is prime, then p = 2 or q = 2
Proposition 2.2.3. For all n ∈...

Prove that n − 1 and 2n − 1 are relatively prime, for all
integers n > 1.

Prove that if x+ \frac{1}{x} is integer then x^n+ \frac{1}{x^n}
is also integer for any positive integer n.
KEY NOTE: PROVE BY INDUCTION

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