Prove that there is no rational number x such that x2= 3.

Q: Let a, n E N. Assume that n > 2. Show that if Va is rational, then it is an integer.

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Q: Suppose the function f is defined on a measurable set E and has the property that {x ∈ E | f(x) > c}…

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Q: Prove that f (x) = x5 + 2x3 + 4x − 12 has precisely one real root.

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Q: 13. Prove by Math Induction ex > 1+x for all positive integers x > 1

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Q: Question 3 Prove that there does not exist a rational number r such that r2 = 3.

A: Given The rational number r does not exist such that r2=3.

Q: & Give an example of a polynomial (x) € Z[] and a prime p such that f(x) is reducible in Qx] but…

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Q: Suppose that f(x)=xn+an-1xn-1+...+a0 is an element in Z[x].  If r is rational and x-r divides f(x),…

A: It is given that, x-r|fx. Implies that, r is a zero of f(x). Since r∈Q, let r=pq,p,q=1,p,q∈Z. Now,…

Q: 1. Prove the following. i. If a ER is such that 0 0. then a = = 0. ii. There does not exist a…

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Q: Prove that for all positive integers n.

A: To prove for all positive integer n ∫01(lnx)ndx=(-1)nn!

Q: Let X be the set of real numbers in the interval [0, 1] with infinite decimal expansions containing…

A: In real numbers a set A is compact if and only if A is closed and bounded in real number system.…

Q: Prove that if x and y are rational numbers such that x > y, then there exists a rational number z…

A: We have to prove that If x and y are rational numbers such that x > y Then there exists a…

Q: Prove f:N - N. F(X) = 5x is injoctive.

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Q: Prove that f (x) = x3 + 3x2 + 6x has precisely one real root.

A: Number of real zeros of a quadratic equation can be determined using its discriminant. For a…

Q: Let f(x) be defined for all rational number a in the interval [0, 1]. Let F(x) and F;(x) be defined…

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Q: Use the Factor Theorem to prove that x + c is a factor ofxn + cn if n Ú 1 is an odd integer.

A: To prove the given statement.

Q: Prove that x2 + 3x + 3 > 0 for all x.

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Q: Show that there are an infinite number of rational numbers between any two real numbers x and y with…

A: using Archimedean property to prove above therom...

Q: Show that for every positive integer n there exists an irreducible polynomial of degree n over Fp.

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Q: Show that if fi(x) is O(g1(x)), f2(x) is O(g2(x)), and f2(x) # 0 and g2(x) # 0 for all real numbers…

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Q: prove for any 2 non negative integers x and y, where x>y, if their gcd IS Y, THEN THEIR LCM is x

A: The objective is to show that LCM(x,y)=x, given GCD(x,y)=y; where x and y are non-negative integers…

Q: At least two of the zeros of f(x) = 9x + 27x – 22x? – 38x + 84 are rational. Find all of the zeros…

A: We have to find

Q: Prove that x2 + 3x + 3 ≥ 0 for all x.

A: We have been given the quadratic equation  x2+3x+3

Q: Prove that for all real numbers Ex)y= -(1u)

A: We have to prove that for all real numbers (- x)y= - (xy)

Q: Verify that the hypotheses of Rolle’s Theorem are satisfied for f(x)=x^2-14x+45 on the interval…

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Q: Prove that there exists a unique positive real number, x, such that x = x^2.

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Q: Prove by reductio ad absurdum that there does not exist a minimum rational number greater than zero.

A: The proof by method of reductio and absurdum mean proof by contradiction . That is , you have a…

Q: 1. Show that 2x2-K = - (2K-1D has rationgi roots for all rational values of K.

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Q: Suppose f (x) is a polynomial with odd integer coefficients and evendegree. Prove that f (x) has no…

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Q: 43. Show that there cannot exist rational numbers a, b such that V3 = a+b/2.

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Q: Prove that if x is rational and x = 0, then 1/x is rational.

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Q: prove for every positive real number x that 1+1/x2>=1/x+1/x3.

A: To prove: 1+1x2≥1x+1x3; for everypositive real number x.

Q: 1. Prove the following. i. If a ER is such that 0 0. then a = 0. ii. There does not exist a…

A: 1)      We will prove it by contradictionLet a≠0  and a>0 (cannot take -ve)now, ε>0 and we can…

Q: Give the smallest extension of Q that contains all the zeros of (x4 – 8x² + 15)(x² + 1). -

A: Yes

Q: Demonstrate that x' + 3x – 8 is irreducible over Q. -

A: Given function is fx=x3+3x2−8 We have to show that fx=x3+3x2−8 is irreducible over Q(Rational…

Q: There exists two odd integers whose average is odd. 1 1 1 There exists distinct positive integers a,…

A: (a) Consider two odd integers 2m+1, 2n+1 such that m+n is even. Now, Consider the average of the…

Q: Suppose that f(x) = xn + a(n-1)x(n-1)+...+ a0  is an element of Z[x]. If r is rational and x - r…

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Q: Given a function f: R → P(Q), where R is the set of real numbers and P(Q) is the power set of…

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Q: Let K be a finite extension of Q. Show that K contains only finitely many roots of unity.

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Q: Suppose f(x) is a degree-2 polynomial. How many solutions x are there to f(x) = 0 (mod 31 - 89) such…

A: Given: In case f(x) is a degree-2 polynomial  f(x) ≡ 0 (mod 31.89) such that 0 ≤x≤31.89

Q: Prove: There exists a positive real number z such that z2=2.

A: Any complex number can be considered as z=x+iy where x,y∈ℝ . Note some result : If z is real number…

Q: Find all positive rational numbers x and y for which x + and y + - are natural numbers.

A: Natural numbers consists of the set {1,2,3,...}. A positive rational number is in the interval…

Q: Show that V3 € Q[6]

A: Q is the field of rational numbers such that…

Q: (a) Show that for x > 0, 1 5a' < In(1+ x) < |

A: As per guidelines, we will solve only first part. Kindly repost other part separately.

Q: Prove that the product of all the distinct nth roots of 1 is ( - l)n+ 1

A: Let us complex number zz=(1)1n apply power n on both sidezn=1zn-1=0

Q: Prove that the product xy of two real numbers x and y is nonnegative if if and only if the absolute…

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Q: prove

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Q: Prove that if y = f(x) is continuous and f(x) = 0 when x is a rational number, then f(x) = 0 for all…

A: Here, we have been given y = f(x) is continuous and f(x) = 0 when x is a rational number. And we…

Q: 4. Verify thatf (x) = x³ – x2 – 20x + 6 satisfies the three conditions of Rolle's Theorem on [0,5].…

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