5. By considering Z[¹+√5]*, or otherwise, show that if n and m are expressible as 2 a² + ab-b² and c² + cd-d², respectively, then nm is expressible as e²+ ef -f² for some integerse and f.

Q: for any integer a, show that gcd(2a+1,9a+4)=1.

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Q: (4) Evaluate 1 z² + 4z - 5 J -dz.

A:

Q: Evaluate xdx A√²+x+1-³x+2+ √(x + 2)² +²1+C B. √² + x + 1 - ² 1 x + ³ + √(x + 2)² + ² 1 + C…

A: To evaluate the given integral- ∫xdxx2+x+1

Q: Rationalize the numerator of √x -4 √√x √√√x-4 √√x Question 15, 10.5.77 HW Score: 33.33%, 5 of O…

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Q: Determine integers x and y such that (а) gcd(51, 288) — 51х + 288у (b) gcd(357, 629) = 357x + 629y

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Q: 3. Show that Z10 is isomorphic to Z2 × Z5, but that Zg is not isomorphic to Z₂ × Z4.

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Q: 19 (e) a = 1575, b = -231 (a) 3710

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Q: Let a and b be integers so that (a, b) = 1. Prove that (a + b, a – b) divides 2.

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Q: For any non negative integer n , prove that an − bn is divisible by a−b,where a and b are any…

A: answer

Q: 5. For how many different integers bis the polynomial x² + bx + 16 factorable over the integers?

A: From the given data

Q: 17 3. Evaluate: de (r- 1)(2 + 16)

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Q: Which of the following is the correct result of 07 15 02 O 15 2016 -I +C 5 H wj* 05 1 21 :+C +0 of […

A:

Q: 3. Discuss Q[√√2] and Q(√2).

A: Q[2] and Q(2) are both extensions of the rational numbers Q obtained by adjoining the square root of…

Q: how the element 2+√3 has a multiplicative inverse in Q(√3). Show that Q(√3) is a subfield of C

A:

Q: 3 S ₁² √ (²√₁² - ²√+²) ³ dt

A: Given

Q: 11.) What is an equivalent form of 4 5+i -?

A:

Q: If m and n are rational numbers such that √m +√n = √5+√24 ,calcule a possible value for m² + n²

A: Given that √m +√n = √5+√24 To calcule a possible value for m² + n²

Q: Suppose the multiplication cx is defined to produce (cx1, 0) instead of (cx1, cx2 ). With the usual…

A: The multiplication cx where x∈ℝ2 is defined as cx=cx1, 0 because x=x1, x2 where x1 and x2 belong to…

Q: Evaluate [R(5xy + z²)dV, where B = [-10, 10] × [0, 1] × [0, 10]

A:

Q: In Z[2√7], show that N(6 + 2√-7) 5 N(1 + 3√-7) but 6 + 2√-7 and 1 + 3√-7 are not associates.

A: As per the question we are given the extended field ℤ [2√-7] and we have to prove that N(6 + 2√-7) =…

Q: Show that r3 + 3x² – 8 is irreducible over Q.

A: Here Q is a field and f(x)= x³+3x²-8 is of degree 3. Then by ROOT TEST we can say that f is…

Q: ndicate the least integer n for which (3x5 + x2)4 is O(xn).

A: Solved using the binomial theorem

Q: show that there doesnt exist a rational numbert such that t2 = 3

A: According to the given information, it is required to show that there does not exist any rational…

Q: a) 1 (x-1)(x+2)(x+4)

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Q: Show that the polynomial x + x

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Q: 1. Compute the resultant of x – 3x* – 2r + 3x + 7x + 6 and x* +² + 1. Do polynomials have a common…

A: Given: To calculate the resultant of .

Q: answer only 9

A: Step 1: Step 2: Step 3: Step 4:

Q: So which of the following is the equivalent of a? A) 3X - 6 B) (X - 3) / 4 C) 3X + 4 D) (X +…

A: As per Bartleby guidelines we have to solve only one question at a time. given  dividend = X…

Q: S tdt √1+2t

A: To integrate we substitute 1+ 2t = z. Then 2 dt = dz  or, dt = dz/2 and  t = (z - 1)/2 .

Q: Show that √3 € Q [√6].

A: Let 3∈Q6 Then there exists rationals a,b such that 3=a+b6 Consider:…

Q: 1) х*- 27 2 - A) (x – 3) B) (x- 3)(x² + 3x + 9) C) (x+ 3)(x² – 3x + 9) D) (x – 3)(x + 3)

A: Due to Bartleby guidelines, only 1question can be solved. Please repost the remaining questions to…

Q: Prove that n2+n  is  divisible by 2, where nN.

A:

Q: If n = 12 m (me N), prove that "C₂ (2+√3)² "Co- - + "C4 (2+√√3)4 n "C6 (2+√3)6 ·+···=(-

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Q: 6. Which polynomials have 4 roots? Select all that apply. 3x* – 81 = 0 4x + 4x + 4 = 0 2x4 + 3x? +…

A: The root of the polynomial is measured by its degree. Degree of a polynomial is the highest power of…

Q: Which of the following is equal to 1-√3|? 01-√3 O√3-1 O (1-√3)² O√2 O√

A: