i. Let F be a field. Prove that a polynomial p(X) = F[X] of degree ≤ 3 is irreducible if and only if p(X) has no zero in F. ii. Show that the preceding statement is false when deg p(X) = 4.

Q: Let f(x) be a polynomial function whose coefficients areintegers. Suppose that r is a real zero of f…

A: Consider an example:  2x3=x+6x+12x-42x3=x3+14x2-2882x3-x3-14x2+288=0⇒x3-14x2+288=0

Q: Let a E K and let f(x) E K[x]. Prove that the remainder when f(x) is divided by x – a is ƒ(a).…

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Q: 6. Use the fact that every odd degree polynomial with real coefficients has a real root to show that…

A: Show that the function f:ℝ→ℝ defined as fx=x5-2x2+x is onto for all x∈ℝ. Definition:  A function…

Q: Let R[x] be the set of all polynomials with real coefficients. In other words, x] = {anx" + an-1x"-1…

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Q: Let f(x) = Q[x] be an irreducible polynomial and assume that f(x) splits in R. Let y E R be a root…

A: To prove that 2^(2/3) + gamma is a primitive element of Q(gamma, 2^(2/3))/Q, we need to show that it…

Q: (a) Prove that, if a and b are nonnegative integers such that a =4 b, then 3ª = 5 3b. ) Using (a),…

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Q: Let α be a zero of f (x) = x2 + 2x + 2 in some extension field of Z3.Find the other zero of f (x) in…

A: The given polynomial is f(x)=x2+2x+2 and α is one of the roots. To Find: the another root of the…

Q: .3 Let f(x) = Use the Bisection method three times to find three successively smaller intervals…

A: Given: f(x)=-x3+2x-2 Here, f(-4)=54 and f(0)=-2 So, The given function is continuous, and the root…

Q: Let x0, x1, ..., Xn be distinct points on the interval [a, b], where Xo <x₂ < x₂ <--- < x₂ and let f…

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Q: Let p: R → R be a polynomial of odd degrees. Prove that the equation p(x) = 0 has a solution in real…

A: Given p:R→R be a polynomial of odd degree. We have to prove that the equation px=0 has a solution in…

Q: 2. Let char F = 0 and f(x) E F[x] be a monic polynomial having positive degree. If d(x) = gcd(f(x),…

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Q: 3. Let P(x) be a polynomial of degree n. If n is even, prove P(x) cannot have an inverse function.

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Q: Let α be a zero of f (x) = x2 + 2x + 2 in some extension field of Z3.Find the other zero of f (x) in…

A: Given: α is a zero of fx=x2+2x+2 in some extension field of Z3. We have to find the other zero of…

Q: ) Prove that, if a and b are nonnegative integers such that a =4 b, then 3ª =5 3b. Using (a),…

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Q: an

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Q: Let f(x) = -x³+2x-2. A zero of f is contained in the interval[-4,0]. Use the Bisection method three…

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Q: 8. Suppose f is analytic in |z| 12. Hint: Write f as a product of a certain quadratic and another…

A: The function f is analytic in z&lt;3, and f-1=f1=0, f1=0, f0=4To prove: there exists z0 with  z0=2…

Q: Let fx)=x+x +2 be a Polynomial in Z3[x]. Which of the following statements is True O f(x) is…

A: The given polynomial function is fx=x3+x2+2 in ℤ3x. Substitute x=0, 1 and 2 in fx=x3+x2+2 and…

Q: 2 (a) Prove that, if a and b are nonnegative integers such that a 4 b, then 3ª = 3. (b) Using (a),…

A: Given a, b are non negative integers such that a≡4b . Then to show 3a≡53b. To show the function…