a. Is (x - 1) a factor of x" c. Is (x - 1) a factor of x" - 1? e. If n is a positive even integer, is (x – 1) a factor of x" – 1? - 1? b. Is (x + 1) a factor of x' d. Is (x + 1) a factor of x" f. If n is a positive odd integer, is (x + 1) a factor of x - 1? 100 100 - 1? 99

Q: 1. Let y be a positive real number. Prove that for every n e N there exists a unique positive real…

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Q: Prove that if the equationf(x)=a0xn+a1xn−1+…+an−1x=0has a positive root x=x0, then the…

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Q: Let f(x) = x2 + bx + c, b is negative odd integer, f(x) = 0 has two distinct prime number as roots,…

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Q: x - x3 + x² – x + 1 is reducible over Q True False

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Q: Let X = {x E Z|x = Y = {y € Z]y = 3n for some integer n}. 6m for some integer m} and let %D a. Prove…

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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: Find k such that f(x) = x° - kx? + kx + 35 has the factor x - 5.

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Q: Show that for all x>0, e* >1+x+÷ 2.

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Q: Find the value of k such that A is singular 1 0 5 4 -1 0 L6 5 k k=

A: given isA=1054-1065kmatrix A is singular if |A|=01054-1065k=01(-k-0)-(4k-0)+5(20+6)=0

Q: Which of the following is homogenous of smallest degree? (A) In x - In y B x+y x-y C √x-y x² + y4 X…

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Q: If f (x) = 3x ^ 4 + 5x ^ 3 + x ^ 2-2 then write f (x) as the product of irreducible terms of Q

A: We have to express f(x) as the product of irreducible terms of Q.

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: ·C) - G) + G)-G) + +-1 - Prove that for any positive integer n, 0.

A: Use simple property of combination.

Q: The characteristic of Z6 x Z, is 18.

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Q: 4. Prove that if f(x) = aox" + a,x"-1+ ...+ an-1x = 0 has a positive root x = xo, then the equation…

A: Roll's theorem: If f is continuous function on [a, b], differentiable on (a, b) and f(a) = f(b) then…

Q: Let f(x) = x3+x +4 and g(x) = 2x +3 in Z5[x]. Then, upon dividing f(x) by g(x), we get O The…

A: *concept:- perform synthetic division.

Q: 6. Suppose that f(x) = x" + an-1r*-1+ ...+ ao E Z[x]. Ifris rational and x –r divides f(x), prove…

A: We will use division algorithm to prove the fact.

Q: f x and y are two positive real numbers, then there exists a positive integer n such that ny>x

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Q: Let f(x) = x³ + x + 4 and g(x) = 3x + 2 in Z5[x]. Then, upon dividing f(x) by g(x), we get The…

A: Elements in Z5 are Z5={0,1,2,3,4}.

Q: Find all values of x such that x² 2+ x + 2! = 0. 3! 4! If no such values of x exist, prove this. + +

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Q: a) There is a unique [x] E Z6 such that [5] - [r] = [2] b) There is no [x] € Z6 such that [3] - [r]…

A: We use the multiplication table for Z6 to verify the given statements:

Q: 3. Determine the pole and the residue of 22+1 f(x) 24-z-2

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Q: (b) Expand and simplify in ascending powers of x as far as the term in x" 1+3x

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Q: (b) Let g(x)=√x with domain {x|x>0}, and let e > 0 be given. For each c > 0, show that there exists…

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Q: ve that x is odd if and only if 3x +1 is even.

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Q: 12. Start with the identities x2 1 P, (x) = xPn-1(x) + *P(x) n P, (x) = xP-1(x) + nP,-1(x ) (a) Show…

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Q: If 4x³ + 9x² - 1 is divided by 2x+1, then the restriction on x is OA) x = 1/2 Q¹ B) x1 O c) * = -…

A: Given: Multiple choice questions. Question 1) If 4x3+9x2-1 is divided by 2x+1, then restriction of x…

Q: Find the value(s) of k such that A is singular.

A: A matrix is singular if its determinant value is zero. So we find the determinant value of the given…

Q: Let x be defined as x = x² -x + 1 for all values of x. If p = p -3, what is the value of p? A 1 BO…

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Q: It can be shown that for any a > 0, there is a r such that at=ert for all real values of t. Find…

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Q: Show that f (x) = 3x² - x - 1 is O(x²) with suitable C and k.

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Q: 3. For f (x) = 2x³ + 4x² + 3x – 7, prove that there exist a number k such that f(k) = 20.

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Q: if x=-2 is a zero of P(X) and that zero has multiplicity 3, then a factor of P(x). is:

A: if x=-2 is a zero of P(x)So (x+2) is a factor of P(x).

Q: 3 6. Let A= 0 x+1 1 x -8 x-1 For what values of x does A-¹ exist?

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Q: Prove that: (1+ x)*** (n+ 1)x C. + + 2 3 n+1

A: I am going to solve the given problem by using some simple calculus to get the required result.

Q: 10-If p is a prime number, then in Z[x] the polynomial x -1 –1 = a) (x - 1) (x + 2) (x –- 3)... (x +…

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Q: Let f(x) = x3 + x +4 and g(x) = 2x + 3 in z,[x]. Then, upon dividing f(x) by g(x), we get The…

A: Given polynomial is

Q: Calculate the following limits using the factorization formula

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Q: find an integer n such that f(x)= 0 for some x between n and n+1. f(x) = x^3 - x + 3

A: Given:The function f(x) = x3−x+3.Known fact:Newton Rphson's method.It is an iterative process to…

Q: 1 and the domain for g(x) is a s as x-> inf, f(x) -> inf and the

A: Domain of a function: All the possible values of x that makes the function f(x) valid or the values…

Q: Let A = {4n^2 + 1 | n ∈ Z}. Show that the size of A is equal to the size of Z≥0. Z≥0 is the set of…

A: A set A is said to be countable  if there is a bijective map between A and set of natural numbers .…

Q: Prove: for n > 1, F; – Fn+1 Fn-1 = (-1)".

A: F_n are the Fibonacci numbers which are defined by F_0=1,F_1=1 and F_n = F_{n-1}+F_{n-2} for…

Q: Find the value of k such that A is singular. 0 3 2 -1 0 L 3 3 k k =

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Q: Let f(x) = x3 + x +4 and g(x) = 2x + 3 in z,[x]. Then, upon dividing f(x) by g(x), we get The…

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Q: (b) For all positive integers n, show that the value of un is the sum of the primitive nth roots of…

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Q: Find the value of k such that A is singular. 0 4 6 -1 0 4 2 k k =

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Q: Let f(x) = x³ + x + 4 and g(x) = 3x + 2 in Zs[x]. Then, upon dividing f(x) by g(x), we get The…

A: See the detailed solution below.

Q: Let A={(-1)" Find sup A and inf A. Provide some explanations how you got their values or why they…

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Q: Let x +1 be a real number. Prove that for all n E N, x" – 1 - + x" +...+ x2 + x + 1. x – 1

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Q: Given the function h(x) = In(-x² + lx) with k a real number and k > 1. If h'(x) - -2x+k lx - x? %3D…

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