If you let R=Z[x]and I = (x^2 -2) be the principal ideal generated by f(x) = (x^2 -2). If r=2x + I exists in R/I, How do you prove that r^2=8 + I?

Q: Assume xo = x2 – 2h, *1 = 12 – h, a3 = x2 + h, 24 = x2 + 2h Use the undetermined coefficients method…

A: To solving the system we use the Gaussian Eliminating and back substitute method

Q: 2. Find the first four Taylor polynomials for, f(x)= In x about x=2

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Q: Which of the following functions are analytic in 1ZK 2 and /or IZ ) > 2: iii) 2-√√24-42² 2²+62+9 ii)…

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Q: A natural cubic spline is defined by Sa,(x-1)' +b,(x-1)² +c, (x - 1) + d, S(x) =- if 0sxs1 (x- 1)' +…

A: Step:-1 Since, we know for cubic spline S(x)=S0x=a0x3+b0x2+c0x3+d0…

Q: Given F(x)=x^2-209 , a=4 ,b=15 Find C1,C2,C3,C4,C5, by bisection method?

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Q: 6. Show that when 0</z - 11 < 2, Complex Analysis (z-1)" (2-1)(2-3)=-3 (2- #=() 2(z-1)

A: Here we prove this statement by using series expansion.

Q: Let 1, Z₁, Z2, Z3, ..., Z-1 be the nth roots of unity. Then prove that (1-z₁)(1-z₂) ... (1-Zn-1) =…

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Q: 3) a) Determine the linear Lagrange interpolation polynomial that passes through the points (-1, 2)…

A: Since you have posted multiple questions, we will provide the solution only to the first question as…

Q: Find the residues of the followings functions at z = 0: (1) (z² + 1)/z (2) (z² + 3z – 5)/2³ (3) (2z…

A: 3. Given function is 2z+1zz3-5.

Q: Find the residues of the followings functions at z = 0: (1) (z² + 1)/z (2) (z² + 3z – 5)/2³ (3)…

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Q: 2.33 Let p(r) = x4 + x³ -- 2x2 + x + 1. Prove that the polynomial equation p(x) = 0 has a root in…

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Q: 7. Evaluate * (D5 - 3D4 + 8D³ + 16D² - 9D - 13)y = 0 y = (C₁ + C₂x)e¯x + С3ex +C4e²x cos3x + C5e²x…

A: Since the student has posted multiple questions and does not mention any specific question, so we…

Q: Evaluate the Cauchy principal value of 2 x 50% -dx 8 x 2 (x² + 1)²

A: We have to find Cauchy Principal Value of the given integral.

Q: Find the residues of the followings functions at z = 0: (1) (z² + 1)/z (2) (z² + 3z – 5)/2³ (3)…

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Q: Please help me prove this: Suppose x, y ∈ R. Prove that if 3x^3 + 21y = 3y^3+ 21x, then x = y or…

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Q: Show that the characteristic of Z[i]/&lt;a + bi&gt; divides a2 +b2.

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Q: 7. Find a pair of integers (x,y) such that 4021x + 2014y= 1.

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Q: Verify that u(x, y) = x² − 6x²y²+y¹ is harmonic everywhere and determine its harmonic conjugate.

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Q: If x,y,z are real numbers satisfying the equation 25 (9x* +y° )+9z²-15(5xy+yz+ 3zx )=0 then prove…

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Q: Suppose f(z) = u(x, y) is a real valued function on an open set U which we are viewing as a…

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Q: Let S,(x) = 2x³ – 3x² + 3x – 4, 0<x < 1, S,(x) = x³ + b,x² + c,x + d,1 sxs 2. Find b,, C1, d, that…

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Q: 2X+1 1) y-vX: zx+\ X { X-4

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

Q: Find the GCD(f, g), where f=5x^4+6x^3+4x^2+5x+1, g=6x^3+3x^2+5x in Z7[x]

A: Find the , where ,  in .

Q: If you let R=Z[x] and I = (x^2 -2) be the principal ideal generated by f(x) = (x^2 -2). If r=(2x +…

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Q: Compute the 1st and 2nd degree Taylor polynomials L(r, y) and Q(r, y) for f(1, y) = In(r +y?) at…

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Q: Which of the following is indeterminate at x = 1? x2 +1 х2 — 1 x² – 1 x² +1 х — 1" x +2 Vx +3 – 2'…

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Q: Find the residues of the followings functions at z = 0: (1) (z² + 1)/z (2) (2² + 3z – 5)/z3 (3) (2z…

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Q: Consider the following bijections on R: f (x) = - g(x) = x – 7 Then (f o g)¯'(x) =

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Q: Show that f(x)= x14 – 6x + 75 is irreducible over Q.

A: Eisenstein's Irreducibility Criterion: Let us consider polynomial…

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Q: Suppose a functions satisfies f(5) = 2, f'(5) = 5, f"(5) = -7, and f"(5) = 12. What is the third…

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Q: 17. Use Theorem 4.7 to prove that Vx + 3 2 V3 + x/4 for

A: Let,

Q: If Tn,0(x) = ao +a¡x+ a2x² +.. he function f(x) centered at x = %3D -) True p) False

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