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

Q: The number of irreducible monic polynomials of degree 2 over Z5

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Q: Let T(x) denote the Chebyshev polynomial of degree k. Express the composite polynomial Tn(Tm(x)) of…

A: The Chebyshev polynomial of first kind defined as  Tmx=2xTm-1x-Tm-2x, with initial conditions given…

Q: Use Eisenstein criterion to verify that, if p =2 , f(x)= x2 -2 is irreducible over Q, state the…

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Q: Abstract Algebra

A: To find the smallest degree polynomial f(x) with coefficients in Q, such that f(√2+√3)=0.

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

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Q: i. Suppose p is an irreducible polynomial of degree n with coefficients in Q. Then p has

A: Determine whether p(x) = x3+x+1 is irreducible over  Sol: we know that , let f(x) be a polynomial in…

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: 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: For any n > 1, prove that the irreducible factorization over Z ofxn-1 + xn-2 + . . . +x +1 is π…

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Q: Show that the polynomial 1+x+x² +...+xP-1 . where p is a prime number, is irreducible over the field…

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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 number of polynomials of degree 3 in Z5[x] is: O (4^4)*5 O 5^4 O 4^4 O 4*(5^3)

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Q: e of all polynomials in C[0, ed in C[0, 1] with respect to

A: To prove a subset W is a subspace of V, it must satisfy 3 conditions: 1) 0 ∊ W. 2) closed under…

Q: 10. Decompose x* + 4.x² + 1 into a product of irreducible polynomials in the following rings: (a)…

A: To find - Decompose x4 + 4x2 + 1 into a product of irreducible polynomials in the following rings :…

Q: Determine the minimal polynomial of 2/3 – V5 ov over 0.

A: That's easy. Have a great day!!!

Q: 2. The Maclaurin polynomials for In(1 +x) are x2 Tn(x) = +... +(-1)n-1ª" x – 3 Use this to find an n…

A: The Maclaurin polynomial is an expansion of a function. This is used to estimate the value of the…

Q: If a polynomial is reducible over Q then it is reducible over Z. O True O False

A: To show whether the given sentence is true or false.

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: Use the third Taylor polynomial P3 (x) for f(x) = x² + Inæ about o = O 1 000 1 and evaluate f(2).

A: Given that fx=x2+lnx. We have to evaluate f2 by using the third Taylor polynomial P3x for fx=x2+lnx…

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

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Q: Prove that f(x) is reducible over Z3. Write f(x) as a product of irreducible factors.

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Q: (2) Construct the Simpson 3/8-th formula to approximate f(x)dr by using La- grange interpolating…

A: Given: ∫abf(x) dx and the function f(x), where the interval of integration a, b is divided into n=3…

Q: Verify that the Legendre polynomial Pn(x) satisfiesthe second-order equation(1 − x2)y'' − 2xy' + n(n…

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Q: Prove that the polynomial f(x) 2an +a2", n € N, a E R \ {0} is not divisible by g(z) = r+a

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

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Q: consider the function y=ax2+bx+c, where a is in the set {n∈ℕ|(n|15)}, b is an integer less than 10,…

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Q: Find all monic irreducible polynomials of degree 2 over Z

A: A polynomial of degree 2 in Z3 [x] is irreducible if and only if it has no roots in Z3.

Q: Show that the following polynomials are irreducible over Z[r]. fi(x) = 17r³ + 7x + 3, f2(r) = 2r* +…

A: Using the result: A polynomial of degree 2 or 3 is reducible over the field F if and only if it has…

Q: Let n EZ. Show that f(x)= x³ – nx +2 is irreducible over Q for n+-1,3,5. 72

A: Given: fx=x3-nx+2, n∈ℤ. To show fx is irreducible over ℚ for n≠-1,3,5.

Q: 26. Determine whether the polynomial in Z[x] satisfies an Eisenstein Criterion for irreducibility…

A: see below the answer with explanation

Q: Use Eisenstein criterion to verify that, if p =2, f(x) =x2-2 is iireducible over rational numbers,…

A: Eisenstein's criterion states that if Q(x)=anxn+......+a1x+a0, a polynomial with integer…

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

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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: polynomial in V2+i and compute the conjugates of i over Express i as a Q(/2 + i).

A: As per the question we have to express i as a polynomial in (√2 + i) and then compute its conjugate…

Q: ow that the polynomial x° + x³ +1 is irreducible over the field of rationals Q.

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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: consider the function y=ax2+bx+c, where a is in the set {n∈ℕ|15}, b is an integer less than 10, and…

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Q: Find two distinct cubic polynomials over Z2 that determine the same function from Z2 to Z2.

A: Given two distinct cubic polynomials over Z2 that determine the same function from Z2 to Z2.

Q: Let f (x) = x + ax +b. How many reducible polynomials of f(x) over Z, are there?

A: Given, fx=x2+ax+b, we have to find the number of the reducible polynomial of f(x) over Z2.

Q: Let a be the positive real fourth root of 3. Factor the polynomial x4 – 3 into irreducible factors…

A: x4-3 =0x=31/4, -31/4,31/4i,-31/4i these all are irreducible with respect to Q…

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: (1) Show that there are infinitely many irreducible polynomials in F[].

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Q: Show that the polynomial x® + x³ +1 is irreducible over the field of rationals Q.

A: The given polynomial is x6+x3+1.

Q: What is the field of fractions of Z[x], the ring of polynomials with integer coefficients?

A: Please check the answer in next step

Q: Find all monic irreducible polynomials of degree 2 over Z3.

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Q: Let a E Z2 be a zero of the irreducible polynomial x3 + x +1 over Z2. Show that the elements of B =…

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Q: Use the error bound for SN to obtain another proof that Simpson’s Rule is exact for all cubic…

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Q: Show that The Hermite polynomials can be difined by the integral representation H, (x) = (-i)"2"e**…

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Q: 5. Show that there are infinitely many irreducible polynomials in F,[X].

A: By the method of contradiction i have solved the problem

Compound Probability

Compound probability can be defined as the probability of the two events which are independent. It can be defined as the multiplication of the probability of two events that are not dependent. 

Tree diagram

Probability theory is a branch of mathematics that deals with the subject of probability. Although there are many different concepts of probability, probability theory expresses the definition mathematically through a series of axioms. Usually, these axioms express probability in terms of a probability space, which assigns a measure with values ranging from 0 to 1 to a set of outcomes known as the sample space. An event is a subset of these outcomes that is described. 

Conditional Probability

By definition, the term probability is expressed as a part of mathematics where the chance of an event that may either occur or not is evaluated and expressed in numerical terms. The range of the value within which probability can be expressed is between 0 and 1. The higher the chance of an event occurring, the closer is its value to be 1. If the probability of an event is 1, it means that the event will happen under all considered circumstances. Similarly, if the probability is exactly 0, then no matter the situation, the event will never occur.