Among the fields C, RQ, Z/2, Z/3, indicate all fields F such that the polynomial X? – X +1€ F[X] is irreducible. Select all that apply: O Field C O Field R Field Q Field Z/2 Field Z/3

Q: for any integer a, show that gcd(2a+1,9a+4)=1.

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Q: 2. Let P(x) = ax² + bx + c be an irreducible polynomial in Q[x). Prove that (a) P(x) has two…

A: Let P(x)=ax2+bx+c be an irreducible polynomial in Q[x] a) To show P(x) has two distinct roots, let…

Q: 2х1 + 3х, — хз — 2 Зх1 — 2х2 + хз — —1 = -1 — 5х1 — 4х2 + 2xз — 3

A: Given that  The system of equations  To find the solution of system of equations by using cramers…

Q: Express the polynomial a+ 3 +a2 + 2r as a product of its irreducible factors in Z,lr).

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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…

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Q: Prove that ithe polynomial PCO =-2xP +2x2+loxx t6 has and fachor azero at M= 3, and fachm s ala the…

A: The given polynomial is px=-2x3+2x2+10x+6. It is given that x=3 is a root of the polynomial. If x=3…

Q: 1. (10 points) Consider the set V of polynomials of the form ac® + bx4+ca² +d, where a, b, c, d e R.…

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Q: Given the polynomial f(x) = -2x7 + 4x³ + 4.x³ following) а) 10 determine the end behavior of f…

A: see the attached file for a step-by-step explanation of end of the behavior of a function 'f'.

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: W laLer Prove the polynomial identity below: (2 – 4y² )² + (4zy)² = (z² + 4y)

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Q: Find a monic, irreducible polynomial + a6 X6 + a5 X$ + a4X4 + a3X³ + azX² + ajX' + ao E Q[ X] ...…

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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 real analysis,Prove that if A is countable, then m(A) =0.

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Q: Evaluate the following commutators [Â, B] = ÂB – BÂ. Hint: use an arbitrary function that cancels at…

A: a x,PxPx=-ih∂∂xx,Pxφ=x,-ih∂∂xφ=-ihx∂φ∂x+ih∂∂xxφ=-ihx∂φ∂x+ih∂φ∂x+ihφ=ihφ∴x,Px=ih

Q: Use Lagrange interpolating polynomials of order 4 to calculate (7.1) 1 3 8 Ax) 1 20 94 267

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Q: For all z € C, let P(z) = ao+a₁2+...+anz" be a polynomial, where a, &C for all j= 0,..., n. Suppose…

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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: Let q represent the degree of the following polynomial 5xym -43y'z'+21. Find the value of q- 2. 19…

A: If a polynomial contains more than one variables then we must add the powers of each variables of…

Q: 2х1 + 3x2 — хҙ — 2 Зx1 — 2х2 + Xз — —1 -1 %3D -5x1 4x2 + 2x3 = 3

A: Consider the provided equations, 2x1+3x2-x3=23x1-2x2+x3=-1-5x1-4x2+2x3=3 Solve the above equation by…

Q: Consider the homogenous polynomial ai + 10r1r2 + x². Find a suitable substitution s = Question 3 azi…

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Q: 15) -3 + V2, -1+ Vi A) -3 - V2, -1 - VT B) 3+ V2, 1+ vT C) None DI 3

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Q: Whether the polynomial x² + 2x + 2 over Z3, Z5, and Z7 is irreducible or not and state all zeros.

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Q: Show that i e Q (/2 ). .

A: Show that i∉ℚ2.

Q: For an integer t, prove that gcd(2t + 1, 9t + 4) = 1.

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Q: Given a positive integer d, define Q[vā] = {s(Va) : f € Q-]}. i.e. the set of real numbers that can…

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Q: Q2) Determine whether each of the following polynomials is irreducible over GF(2). Justify your…

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Q: Check whether i e Q(v-2) or not.

A: Let p,q ∈Q such thatp2-2q2+i22pq=p+q-22=-1then either p=0 or q=0, as22pq=0 Since -1 has no imaginary…

Q: Under what conditions (if any) are polynomials (over R) injective? Surjective? Bijective? Explain…

A: Let us consider the n degree polynomial px=anxn+an−1xn−1+......+a1x+a0. Here, an≠0 and x∈R…

Q: Assume that (8) ∞∑n=1nanxn−1=∞∑n=0anxn∑n=1∞nanxn−1=∑n=0∞anxn for all x, and determine what this…

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Q: If F = Za[z] (z²–cz+1) find the value of c for which F will be a field? %3D

A: F is field iff x2-cx+1 is irreducible over f.

Q: Which of the following polynomials is reducible over Q: x3 +2 x4 + 3 x5 + 1 x2 + 3 I thought it…

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Q: 1) f(x,y)= 2x² – 3y – 4 2) f(x, y) = (x² – 1)Xy + 2)

A: Solution :-