Question

**NUMBER THEORY**

Proof that the following polynomials do not have integer roots.

a) x^{3} − x + 1

b) x^{3} + x^{2} − x + 1

Answer #1

**NUMBER THEORY**
Proof that the following polynomials do not have integer
roots.
a) x3 + x2 − x + 3
b) x5 − x2 + x − 3.

Find the GCD d(x) of the following pairs of polynomials
a(x),b(x) in Z3[x]. In each case,
express d(x) as a linear combination of a(x) and b(x).
a) a(x)= x3 + x2 +1 , b(x)=
x2+1
b) a(x)= x3 + x2+ 1 , b(x)= x2
+ x + 1
c) a(x)= x3 +x2 +1 ,b(x)= x2 +
x

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...

Find the number of integer solutions to x1+x2+x3=20 given the
following restrictions:
(A) x1>=3, x2>=2,x3>=5
(B) x1>=0, x2>=0, x3<=6

2. Find the number of integer solutions to x1 + x2 + x3 + x4 +
x5 = 50, x1 ≥ −3, x2 ≥ 0, x3 ≥ 4, x4 ≥ 2, x5 ≥ 12.

Determine the third and fourth Taylor polynomials of
x3 + 3x - 1 at x = -1.
p3(x) =
p4(x) =

1) Determine whether x3 is O(g(x)) for the following:
a. g(x) = x2 + x3
b. g(x) = x2 + x4
c. g(x) = x3 / 2 2)
Show that each of these pairs of functions are of the same
order:
a. 3x + 7, x
b. 2x2 + x - 7, x2

How many integer solutions are there to
x1+x2+x3+x4= 100 with
all of the following constraints:
10 ≤ x1 , 0≤ x2 < 20 , 0 ≤
x3 < 40 , 10 ≤ x4< 50

Apply Gram-Schmidt in L2[−1,1] to the list of functions
1,x,x2,x3. (You do not have to normalize.)

You’re the grader. To each “Proof”, assign one of the following
grades:
• A (correct), if the claim and proof are correct, even if the
proof is not the simplest, or the proof you would have given.
• C (partially correct), if the claim is correct and the proof
is largely a correct claim, but contains one or two incorrect
statements or justications.
• F (failure), if the claim is incorrect, the main idea of the
proof is incorrect, or...

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 1 minute ago

asked 3 minutes ago

asked 18 minutes ago

asked 24 minutes ago

asked 31 minutes ago

asked 34 minutes ago

asked 41 minutes ago

asked 49 minutes ago

asked 57 minutes ago

asked 59 minutes ago

asked 1 hour ago

asked 1 hour ago