6- Show that the Polynomial has a Z4 [x] 2X+1 in multiplicative invers. in 24 [x].

Q: Show that x = 1 is the only zero of the polynomial x^25 − 1 over Z_37

A: The elements of the sets Z37 are {0,1,2,3,.......,36}

Q: The generally polynomials are of the form Select one: a. b. O C. d. an" + an-x-1 + ... + aix+ ao…

A: Polynomial A polynomial is  an expression which includes variables, constants and exponents, that…

Q: Which of the following polynomials is irreducible over Q? * O x^4+x+1 3x 3-6x 2+x-2 O x*2+5x-6 ONone…

A:

Q: Using Legendre polynomials, we can express f(I) = z' as %3D

A: The Legendre polynomials are as follows.…

Q: 16. Find a fourth degree polynomial that has the zeros 3, -2, i.

A: Given data: The first root of the polynomial is x1=3. The second root of the polynomial is x2=-2.…

Q: In Z3[x], show that the distinct polynomials x4 + x and x2 + xdeterminethe same function from Z3 to…

A: We know that ℤ3 is a field i.e., a commutative ring with unity in which every non-zero element has a…

Q: Show that 4x2 + 6x + 3 is a unit in Z8[x].

A: We have to show that 4x2+6x+3 is a unit in ℤ8x.

Q: . Factor x12 -1 as a product of irreducible polynomials over Z.

A: Given: x12-1. We have to factor x12-1 as a product of irreducible polynomials over Z.

Q: Which of the following polynomials is reducible over Q? * x^4+x+1 O 3x^2+x+2 3x^3-6x^2+x-2 None of…

A:

Q: Suppose that x and y are real numbers such that y is 6 greater than x. What is the smallest possible…

A:

Q: Determine which of the polynomials below is (are) irreducibleover Q.

A: According to our guidelines we can answer only one question and rest can be reposted.

Q: In Q[r]/(x² - 4) give an example of a zero divisor. Justify.

A:

Q: Suppose A=| use Diagonalization to compute A" 10

A: In the given question we have to find the 10th power of the matrix A s.t A=10-12. Concept: If a…

Q: product of irreducible polynomials.

A:

Q: If p(x) is a polynomial in Zp[x] with no multiplezeros, show that p(x) divides xp^n-x for some n.

A: Px is a polynomial  Px=x-ax-bx-c...x-ux-vx-w where   a,b,c...∈Zp but u,v,w∉Zp Let Rx=x-ux-vx-w has…

Q: Express x³ + 4x4 + 4x³ −x² − 4x + 1 € Z₁[x] as a product of irreducible polynomials over Z5. Be sure…

A: We have to express polynomial in Z5 given as  x5+4x4+4x3-x2-4x+1  As a product of irreducible…

Q: Give all of the zeros of P(x)=x^2+4 as a comma separated list.

A:

Q: Calculate the commutators [p(hat)x3, x(hat)].

A: We have to calculate the following commutators. p^x3 , x^ Introduction: Commutation of operator A^…

Q: 4. Write the polynomial in standard form. * 4g – g° + 3g² – 2 -2 +4g + 3g – g g - 3g +4g - 2 O A -g…

A: Standard Form of Polynomial states that the polynomials needs to be written with the exponents in…

Q: Which of the following are NOT polynomials? I.  \sqrt(x-9) II.   x^8−14x+1 III.  1/2⋅x+0.149 IV.…

A:

Q: Show that the polynomial x5 - 6x + 3 over Q is not solvable byradicals.

A:

Q: Show that the polynomial p(x) = -x^3+x^2+1 has a root in [1,2]

A: We can check for the values of p(x) :

Q: Show that the polynomial x + x

A:

Q: Write x3 + 2x + 1 as a product of linear polynomials over some extension field of Z3.

A: Given: x3+2x+1. To Do:We have to write x3+2x+1 as a product of linear polynomials over some…

Q: 1. Compute the resultant of x – 3x* – 2r + 3x + 7x + 6 and x* +² + 1. Do polynomials have a common…

A: Given: To calculate the resultant of .

Q: Express x8 - x as a product of irreducibles over Z2.

A: Given polynomial is fx=x8-x. To Express: The given polynomial fx=x8-x as a product of irreducible…

Q: Write x3 + 3x² + 3x + 4 in Zs[x] as a product of irreducible polynomials.

A:

Q: Which of these polynomials has a zero of multiplicity 3 at z = 1? a. p(r) = xª + 5x³ + 5x² – 5x – 6…

A: If x=a is a zero with multiplicity n of a polynomial, then x-an is a factor of the polynomial. a.…

Q: The product of x* + x3 + 2x? and 2x² + 4x + 6 in z, [x]/ is:

A:

Q: Express x8-x as a product of irreducibles over Z2

A: We have to express x8-x as a product of irreducible over Z2. Definition:- Irreducibility is defined…

Q: Which of the following polynomials is reducible over Q? * O x^4+x+1 O 3x^3-6x^2+x-2 O 3x^2+x+2 O…

A:

Q: E Show that x'+ 3x+ 2 has 4 zerosin Ze

A:

Q: Determine the number a in Z5 so that (x - a) becomes a divisor of f (x) = x ^ 3 + 2x ^ 2 + 3x + 3 in…

A: