Show that ZV-6 is not a UFD by factoring 10 in two ways. Why does this prove that Z[V-6] is not a PID?

Q: Let p(x) = x³ – 2x + 4 and compute p(A). 1 A = 0 -4 4 NOTE: Write the elements of the matrir…

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Q: Let p be a prime and let q = p². Show that Z,[x] has a unit that is not a constant polynomial.

A: Given that, Let p be a prime and let q = p ^ 2 . Show that Zq[x] has a unit that is not a constant…

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: Use the inner product = aobo+ aıb1+ azbz+ asb; to find , lIlp|], d(p, q), and then find the angle…

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Q: Fit a second-order polynomial to the data in the kollowing daha- 0|12/31415 yi 2.17.713.6|272 40.9…

A: The objective is to find a second order polynomial and to find the standard error.

Q: Show that 2 is equal to the product of a unit and the square of an irreducible in Z[i].

A: We have to show that 2 is equal to the product of a unit and the square of an irreducible in Zi. The…

Q: The characteristic of Z6 x Z, is 18.

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Q: In the field GF(p"), show that for every positive divisor d of n, - x has an irreducible factor over…

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Q: Find the standard inner product P2 of the given polynomials. on p = -3+ 3x + x², q = 2 + 5x – 4x?.…

A: According our guideline i solve only first.For second solution please resubmit next question again.

Q: Find the gcd of (x^4+x^3+x^2+2x+3) and (x^3+4x^2+2x+3) in Z5.

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Q: (2) Find all the numbers z such that elz = 10i.

A: Consider the given. eiz=10i

Q: Find the 2 square roots of 3 - 4j using De Moivre's Therem.

A: Let z=3-4j---(1) By De Moivre's theorem, z=r cos θ + j sin θ ⇒z=rcos θ + j sin θ12 ⇒z=rcos θ2+j sin…

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

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Q: In Exercises 11–12, find the standard inner product on P2 of the given polynomials. 11. p = -2 + x +…

A: We can solve this as follows:

Q: Find the coefficient of at in the product (1 + 2a)* (2 – a) using binomial

A: Introduction :

Q: Show that for all z1, Z2 E C, Then we have: |Iz1| – |z2|| < |z1 +z2l

A: 2) The inequality can be proved as follows:

Q: Show that 2Z ∪ 3Z is not a subring of Z.

A: Theorem for Sebring Test: For any ring R, a subset S of R is a subring if and only if: it is closed…

Q: Identify the splitting field of the polynomial x2 – aª over Q(7*)

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Q: Find (a) ATA and (b) AAT. Show that each of these products is symmetric.

A: a).

Q: 5. Suppose A = PDP¯1, where P is 2 × 2 and D = 9I. Let B = 51 – 3A+ A². Show that B is…

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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: Let

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Q: Factor 19 into irreducible elements in the Gaussian integers, Z[i].

A: We have to factor 19 into irreducible elements in the Gaussian integers, ℤ[i]. Irreducible element :…

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: Consider a LU factorization of -2 1 3 A = 1 4 -2 -5 as A=LU. Then L2,1 and U1,1 are respectively…

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Q: Use the inner product (o, 9) = 2obo +a1b1 +a2b2 to find o, 9), |P||- |||, and d(p, a) for the…

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Q: Give the smallest extension of Q that contains all the zeros of (x4 – 8x² + 15)(x² + 1). -

A: Yes

Q: Use the inner product (p, q) = aobo + ab1 + azb2 to find (p, q), ||P||. ||9||, and d(p, q) for the…

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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: Prove the following statement: If x € Z, shares a common nontrivial factor with n, then a does not…

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Q: Prove that every perfect square is either a multiple of 5 or of the form 5j - 1 or 5j + 1.

A: Fermat's little theorem states that if p is a prime number, then for any integer a, the number ap −…

Q: Which of the following polynomials in Z3 [x] is irreducible? (a) p (x) = x ^ 3 + x + 1 (b) p (x) =…

A: Useful result: If f(x)∈F[x] be a polynomial over a field F of degree two or three. Then f(x) is…

Q: Show that 1 is the only solution of x25 - 1 = 0 in Z37.

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Q: find all real zeros for the given polyn 20x + 12; x + 3

A: Polynomial: f(x)=3x3+x2-20x+12 The zeros means at those values f(x) will be zero.

Q: Find the standard inner product on P of the given polynomials. p= -2+ 4x + a2, q = 3+ 4x 3a2. (p, q)

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Q: if n is a perfect square then n2 + 2n is not a perfect square. prove this

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Q: Find the invariant factors of Zg x Z1, X

A: (c) We have to find the invariant factor Z8×Z12×Z15 First we will find the elementary division…

Q: The entries of A are aij. Assuming that zeros don't appear. what is (a) the first pivot? (b) the…

A: A =aij  where aij≠0we reduce A in row reduced echelon form first pivot  element is a11 bacause every…

Q: Consider a LU factorization of 3 A = 6 as A=LU. Then L2,1 and U1,1 are respectively equal to

A: solution:-

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: Show that 2Z U 3Z is not a subring of Z.

A: show that 2Z∪ 3Z is not a subrings of Z:   Solution:-             Let 2∈ 2Z∪3Z…

Q: The set B = {4+ x2, 12 – 4x + 3x2, 12x - 31 – 8x2} is ab rdinates of p(x) 75 – 24x + 19x? relative…

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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: 6x-C), Prove that If2zf(x+ct)+Ø て

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Q: Find a polynomlal f (x) of degree 3 that has the following zeros. 5, -7, 6 Leave your answer in…

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Q: 2. Determine the greatest common divisor of a(x) = x³ – 2 and b(x) = x + 1 in Q[x] and write it as a…

A: Given the polynomials   ax=x3-2 and bx=x+1 to find gcd(a(x), b(x)) and write it as a linear…

Q: Q3) Use the inner product = a,bo + a,b1 + azb2 to find , llp||, d (p, q), and then find the angle…

A: Given: Inner product: p, q=a0b0+a1b1+a2b2 and the polynomials  p(x)=1-x+x2 and q(x)=x-3x2. To find:…

Q: 10. Suppose that A in has the SVD A= USVT Use this factorization to solve Ar =

A: Note: As per bartleby instruction when more then one question is given only one has to be answered.…

Q: (a) State the Eisenstein Criterion (without proof). [ (b) Prove that 30x³ 91 does not have roots in…

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Q: For all a and b in Z, a - b is a factor of an - bn. (Hint: ak+l - bk+I = ak(a - b ) + (ak - bk)b.)

A: we have to prove that For all a and b in Z, a - b is a factor of an-bn

Addition Rule of Probability

It simply refers to the likelihood of an event taking place whenever the occurrence of an event is uncertain. The probability of a single event can be calculated by dividing the number of successful trials of that event by the total number of trials.

Expected Value

When a large number of trials are performed for any random variable ‘X’, the predicted result is most likely the mean of all the outcomes for the random variable and it is known as expected value also known as expectation. The expected value, also known as the expectation, is denoted by: E(X).

Probability Distributions

Understanding probability is necessary to know the probability distributions. In statistics, probability is how the uncertainty of an event is measured. This event can be anything. The most common examples include tossing a coin, rolling a die, or choosing a card. Each of these events has multiple possibilities. Every such possibility is measured with the help of probability. To be more precise, the probability is used for calculating the occurrence of events that may or may not happen. Probability does not give sure results. Unless the probability of any event is 1, the different outcomes may or may not happen in real life, regardless of how less or how more their probability is.

Basic Probability

The simple definition of probability it is a chance of the occurrence of an event. It is defined in numerical form and the probability value is between 0 to 1. The probability value 0 indicates that there is no chance of that event occurring and the probability value 1 indicates that the event will occur. Sum of the probability value must be 1. The probability value is never a negative number. If it happens, then recheck the calculation.