Question

*Prove that if f*(*x*) =
*a**k**x^k*
+*a**k*−1*x^k+1*
+*a**k*−2*x^k+2*+...+*a*1*x*+*a*0
*is a polynomial in* Q[*x*] *and a**k*
̸= 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.*

Answer #1

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.]

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...

True or False, explain:
1. Any polynomial f in Q[x] with deg(f)=3 and no roots in Q is
irreducible.
2. Any polynomial f in Q[x] with deg(f)-4 and no roots in Q is
irreducible.
3. Zx40 is isomorphic to
Zx5 x Zx8
4. If G is a finite group and H<G, then [G:H] = |G||H|
5. If [G:H]=2, then H is normal in G.
6. If G is a finite group and G<S28, then there is
a subgroup of G...

7. Suppose p(x) = a0 + a1x + a 2 x 2 + · · · + akx k is a
polynomial of degree k. Find the Taylor series of p(x), and find
its radius and interval of convergence.

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
polynomials form an orthonormal set, and if not, apply the
Gram-Schmidt orthonormalization process to form an orthonormal set.
(If the set is orthonormal, enter ORTHONORMAL in both answer
blanks.)
{−2 + x2, −2 + x}
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.) { 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...

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

Q7) Factorise the polynomial f(x) = x3 −
2x2 + 2x − 1 into irreducible polynomials in
Z5[x], i.e. represent f(x) as a product of irreducible
polynomials in Z5[x]. Demonstrate that the polynomials
you obtained are irreducible.
I think i manged to factorise this polynomial.
I found a factor to be 1 so i divided the polynomial by (x-1) as
its a linear factor.
So i get the form
(x3 − 2x2 + 2x − 1) =
(x2-x+1)*(x-1) which is...

For each of the following pairs of polynomials f(x) and g(x),
write f(x) in the form
f(x) = k(x)g(x) + r(x)
with deg(r(x)) < deg(g(x)).
a) f(x) = x^4 + x^3 + x^2 + x + 1 and g(x) = x^2 −
2x + 1.
b) f(x) = x^3 + x^2 + 1 and g(x) = x^2 − 5x + 6.
c) f(x) = x^22 − 1 and g(x) = x^5 − 1.

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