12. Which of the following polynomials is reducible over Q а) 4x3 + x — 2 b) Зх3 — 6х? +х - 2 c) 5x3 + 9x2 – 3 d) None of the above

Q: Prove that: m3 = m – 3m,m, + 2m²

A: Hello! As you have posted 2 different questions, we are answering the first question. In case you…

Q: Now just as with Example 2.16, prove that if f, g, and three polynomials in R(r], then (f+ g) + h =…

A: Introduction: In mathematics, functions are the most fundamental aspect of calculus. Functions are…

Q: Show by any means that Q[r]/(x4 + 3x2 + 9x + 1) is a field.

A:

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: & Give an example of a polynomial (x) € Z[] and a prime p such that f(x) is reducible in Qx] but…

A:

Q: The number of irreducible polynomials over Z13 of the form x + ajr + a2 is 91 O 182 O 78 O 169

A: Given x2+a1x+a2 is a 2nd degree polynomial .For a 2nd degree polymonial , no of irraducible…

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: 3. Prove that {12a + 4b |a, b e Z} = {4c\c € Z}.

A: as per our guidelines we are supposed to answer?️ only one question. Kindly repost other question as…

Q: Show that Q(√7, i) is the splitting field for x4 - 6x2 - 7.

A:

Q: 2. Is the following set of polynomials linearly independent in P2? Explain your work. (-1+2t + 3t?),…

A:

Q: Verify that v2+ V5 is algebraic over Q.

A: Algebraic number :- An algebraic number is a number that is a root of a non-zero polynomial in one…

Q: Consider the following polynomials in P3: a(x) = x³_2x²–2x+3 r(x) = 4x³–x²-3x+6 u(x) = 4x³+9x²+x+3…

A: I am going to solve the given problem by using some simple algebra to get the required result.

Q: Determine which of the polynomials below is (are) irreducible over Q.a. x5 + 9x4 + 12x2 + 6b. x4 + x…

A: According to Eisenstein's criterion, it provides a suitable condition for polynomial be irreducible…

Q: irr(ꭤ,ℚ) and deg(ꭤ,ℚ

A: Given: 2+i To find:  The irr(ꭤ,ℚ) and deg(ꭤ,ℚ)for the given algebraic number ꭤ ∈ ℂ.

Q: Determine the irreducible polynomial for 15 over each of the following (c) a = /3 fields (i). Q (ii)…

A:

Q: Prove that x4 + 15x3 + 7 is irreducible over Q

A:

Q: 18. Let xo < x, < x2, let {yo, y, y2} be given. Find the polynomial q(x) with deg(q) < 2 and such…

A:

Q: If fu)=3Xtl, what is f (5)? 1,4,9,14,25 what ss t he next term in toseauene? W hat is the product af…

A: We have to find f(5), next term and multiple of given term.

Q: Determine if the following polynomials are irreducible or not k(1, y) = 1® – y° € Q!r, y]

A: Given:- k(x ,y)=x6-x7 ∈ Q x,y

Q: Find d = gcd (270, 105) and hence find integers x , y such that d = 270x +105y .

A: GCD stands for greatest common divisor for two or more numbers.

Q: 4.a) Show that (1+2i) is a zero of the polynomial Qcx) = x²-6x*+3x -20. Hence , or otheruwise, write…

A:

Q: Show that any positive integer n with n ≥ 4 can be expressed as n = 2x + 5y where x ≥ 0, y ≥ 0.

A:

Q: Change the following data, such that the data follows a certain fourth-degree polynomial. X: -2 -1 0…

A: Newton's divided difference formula can be defined as:…

Q: Show that Q(√7, i) is the splitting field for x4 - 6x2 -7.

A: x4 - 6x2 -7 = x4 - 7x2 + x2 -7                   = x2( x2 - 7) + 1 ( x2 - 7 )                   =…

Q: Which ring is fx=2x^2-1 irreducible? Z[x] Q[x] Z2[x] Z3[x

A:

Q: For any positive integer n, how many polynomials are there ofdegree n over Z2? How many distinct…

A:

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

A:

Q: 9. Show that 4x - 251 + 10x? – 15x + 20 is irreducible over Q.

A: Answer

Q: Prove that there cannot exist two different polynomials q,r E K, both of degree less than n. such…

A: Given K be any field and a1,a2,...,an be pairwise distinct elements of K. For each i=1,2,3,...,n, pi…

Q: 3. Prove that Re(2-+z')<4 when z<1.

A: a.          x+y ≤x + y                                             (triangle inequality)b. Re(z1+z2)…

Q: Prove directly that no number of the form 4a (8b − 1) can be written as the sum of three squares by…

A: Let us first show that (8b-1) cannot be expressed as a sum of three squares Let us take any integer…

Q: Obtain the Legendre polynomial P4(z) from Rodrigue's formula Pn (2): = 1 d 2"n! dan [(2²-1)"]

A:

Q: Q2: Show that whether (Z16,O.O) is field or not?

A: Properties of a field For all a,b, and c∈F, the following hold: a+(b+c)=(a+b)+cand a⋅(b⋅c)=(a⋅b)⋅c…

Q: By using Eisenstein's criterion we can conclude that the polynomials x^3+4x^2+3 and…

A: Using Eisenstein's criterion we can check a polynomial is reducible or irreducible. For that some…

Q: Demonstrate that x' + 3x – 8 is irreducible over Q. -

A: Given function is fx=x3+3x2−8 We have to show that fx=x3+3x2−8 is irreducible over Q(Rational…

Q: 18. Is Q[x]/(x² – 5x + 6) a field? Why? 10

A:

Q: (c) Show that : f(x) = x2 + x + 2 e Zg[x] %3D is irreducible. Further, find the order of f (2).

A:

Q: Explain why 3Z is not an ideal in Q

A:

Q: Show that the following polynomials are irreducible over Q. (a) f(r) = 13r + 15a + 20a? + 21a + 19…

A:

Q: Consider the following polynomials in 9(x) = -+3-+5 (2) --4+7x+10 u(x) --Sr'+8+1o For each of the…

A: Since you have asked multiple questions, we will solve the first question for you.  If you want any…

Q: 8. give their greatest common divisor. Let F = Z3. Factor the polynomials x³ + x² + 2x + 2 and x³ +…

A: Given the polynomials f(x) = x3+x2+2x+2 and g(x) = x3+1.

Q: Prove that 8x3 - 6x - 1 is irreducible over Q.

A:

Q: 4. Prove that x – 15x³ + 10x² – 21x + 2021 is irreducible in Q[x]. -

A: Given :-  Prove that x5 -15x3 +10x2 -21x +2021 is irreducible in ℚx.

Q: (ii) A = 82 + &V1+ 4

A:

Q: How many irreducible polynomials of degree 4 are there in Z/2[X]?

A:

Q: Prove that the number V7 lies between 1 and 2.

A:

Q: 5. Show that for Chebyshev polynomials (Tn, Tm) = 0, n #m, π, EIN n=m= 0, n = m = 0.

A: We have to solve given problems;

Q: Which of the following polynomials is guaranteed by the Intermediate Value Theorem to have a zero on…

A: Given that, The interval is = [-3, -2] We have to check the functions given in options whether they…

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: 4. Verify thatf (x) = x³ – x2 – 20x + 6 satisfies the three conditions of Rolle's Theorem on [0,5].…

A: