Find the Teylor expansion of 4(2): z" with center 1, where Nis a positive integer. Use the above to prove that; z (2-1)+ 5(2-)*+10(2-1)* \o(2-1)*, 5(2-1)+ L

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A: is a solution to the recurrence, and, for all

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A: We have a group Z4. We have to prove that Z4 is cyclic.

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Q: 10. Factor the polynomial 4x2 – 4x +8 into a product of irreducibles viewing it as an element of the…

A: Factor the polynomial 4x2 - 4x + 8 into a product of irreducibles viewing it as an element of the…

Q: Determine the term involving y5 in the expansion of (3y +).

A: Please refer the attached image for complete solution. THANK YOU.

Q: Prove that if gcd(a, m) =1 and gcd(a – 1, m) =1, then 1 + a + a^2 + ....+ a^ phi(m)-1 congruent to 0…

A: Given gcda,m=1 and gcda−1,m=1 We have to show that 1+a+a2+......+aϕm−1≡0modm Since, gcda,m=1⇒m/a And…

Q: 1. Prove or disprove: The following gives an inner product on R².

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Q: (8) In the ring (Z, +o6), we get n {(a): a prime element in Z,}: (a) o (b) ((0), +66) (c) (Z6, +66)…

A: Prime Ideal: Let R, +, · be a commutative ring. An ideal A of R is said to be a prime ideal of R if…

Q: Prove-that '(2) r' (₂+1) ¹ (2+ |) T(₂) nd thus derive Legendre's duplication formular 2' (2) T(2:)…

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Q: 6. a) Prove that S, is not abelian for all n23. b) Prove that A, is not abelian for all n24. Ay…

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Q: Suppose g has a Laurent expansion o -1) ₂²-(-2) ³ as → 0. 9(2) = Σa,2³, j=-1 valid on a punctured…

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Q: for

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Q: OExpand -7(y – 12)

A: For expand we have to multiply 7 inside. So 7 will be multiply by y and 12.

Q: Suppose that I has minimal lolynomial 2² (2-1) and characteristic Polynomial 24 (2-1)². Write down…

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Q: 25. Which is the correct expansion of (x + 4y)³ using the binomial theorem? O ○ (x + 4y)³ = (3)…

A: Given, (x+4y)3

Q: 8. The Laurent's expansion of (2² – 1) (2² + 4 for |z|<1 is 5z2 2125 1). 4 64 16 2) + 21 4 4 4 3) -…

A: None of the option 1,2,3 and 4 is correct. Please check it.

Q: Find the expansion of fix) = In(1+e"? Seiect one: O si In(2) +z/2+/8+z/192 +. O In(2) + z/2+z²…

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Q: in(Zg++s)indeyclic grauf2enerated by I, Z,3

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Q: Define L: Z →Z and M:Z Z by the rules L(a) = a² and M(a) = a mod 5 for each integer a. %3D (a) Find…

A: Given that L:Z→Z is defined by La=a2 and M:Z→Z is defined by Ma=a mod 5 for each integer a. Now,…

Q: The 4th term in the expansion of (2x -3x 10 is 10 C4 (2a) (-3x)³ O 10 C4 (2a2) (-32)4 10 C (2)…

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Q: 16. Find the term containing x18 in the binomial expansion of (x2- y)15.

A: A binomial expression is an expression of the form a+bn. This binomial expression can be expanded…

Q: (8) In the ring (Z6, +6,6), we get {(a): a prime element in Z,} = (a) ¢ (b) ((0), +6,-6 ) (c) (Z6,…

A: The ring Zn has no prime elements if and only if either n= 1or n is a prime

Q: 12. The first four terms of the binomial expansion of (1 - )' A I+ 4x + 7x² + 7x B 1- 4x – 7x + 7x³…

A: 1

Q: 11. Which is the correct expansion of (x + 4y) using the binomial theorem? O(x+4y)³ = (x) (4y) +…

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Q: Prove that (1,1) is an element of largest order in Zn1 + Zn2. State the general case.…

A: We know that order of any group G×H is lcm(|G|, |H|)     |G| = order of group G. Similar for H. Now,…

Q: 36. Use the binomial theorem to expand (2x - y)* . A. 16x* – 32x³y + 24x²y² – 8xy³ + y* С. 2x - 4х3у…

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Q: Use the Binomial Theorem to expand the binomial (3x – 5y)*. 81x* + 625y* b. 81x* – 540x'y + 1350x²y²…

A: The expansion of 3x-5y4 by the help of Binomial Theorem is

Q: How do you find the term independent of y in the binomial expansions

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