Show that if yª – aynumber and a # 2.= 1 has no rational solution for positive a such that a is a natural

The given equation is ya-ay=1,a≠2 this equation is not valid for a = 1 although it is given for…

Answered: Show that if ya – ay = 1 has no…

Math

Advanced Math

Show that if ya – ay = 1 has no rational solution for positive a such that a is a natural number and a 2.

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter4: Polynomial And Rational Functions

Section4.4: Complex And Rational Zeros Of Polynomials

Problem 38E

See similar textbooks

Related questions

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

Q: Prove that Σ (2n+1) P(x) no (1-2hx+h²) ¾/2₂

Q: 1 1. Solve the rational equation x - x - 6 x- 3 x +2 and state any NPV's.

Q: 1) Prove that there are no positive integer solutions to the equation x2-y?=1

Q: If the function (x) =sin 2(2n+1)x for some positive integer number n, then the conclusion of Rolle's…

A: Given that: f(x)=sin2(2n+1)x on the interval π8, 3π8.and satisfies the Rolle's theorem at…

Q: Let p be a prime and let q = p². Show that Z,[x] has a unit that is not a constant polynomial.

A: Given that, Let p be a prime and let q = p ^ 2 . Show that Zq[x] has a unit that is not a constant…

Q: Let 0 be a root of the polynomial x³ + 3x + 7. Prove that 0 is prime in Q(0).

A: We will use the basic knowledge of polynomials and rings to answer this question properly, in…

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

Q: a. Give the smallest extension of Q that contains all the zeros of x4 - 2x2 – 15.

A: We shall answer first question only as you have asked diff question without mentioning. For others…

Q: (b) Express in rational form the generating function of 2, -2/5, 2/25, -2/125, 2/625,..

A: The given problem is to find the generating function for the given sequence, we can observe same…

Q: Show that x19 + x8+ 1 has multiple zeros in some extension of Z3.

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)

Q: If a, B are the roots of the equation x + px + q =0 and also of x + p"x" + q* = 0 %3! αβ and If are…

Q: Suppose x,y,z are mutually coprime positive integers that solve x2+y2=z2. a) Show why exactly one…

A: Basic properties of modular arithmetic and gcd will be used in solving this problem. The detailed…

Q: Show that f(x) = x4 + 4.x + c has at most two real roots for any umber c.

A: Consider the given equation. fx=x4+4x+c Case 1: When c>0. No sign change, by Descartes rule of…

Q: a) What is the generating function for {ak}, where akis the number of solutions of x1 + x2 + x3 = k…

Q: (9) Is an complex annullus Ann(2o, R1, R2) with 0 < R1 < R2 simple connected?

A: Note: As per bartleby instruction when more then one question is given only one has to be answered.…

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: Let p be a prime. Solve the equation x2 = 1 in Z/p (i.e. find all solutions, and prove that there…

Q: Prove that there are infinitely many solutions in positive integers x, y, and z to the equation x2 +…

A: We already know, (x,y,z) = (3,4,5) is a solution of the equation, x² + y² = z² i.e. 3²+4² = 5² Now,…

Q: The possible integral zeros of P(x) = x ^ 3 + 3x ^ 2 - 4x - 12 are the divisors of _______

A: The given polynomial is:x3+3x2-4x-12 We have to find the divisor of possible integral zeros of P(x)…

Q: 17. Find all roots of the polynomial P(x) = x* – x³ – x² - x– 2.

A: This question is about nth root of polynomial

Q: For any n > 1, prove that the product of the nth roots of unity is(-1)n+1.

Q: (1) Use the definition to prove that if xn → 2 then (2xn + 1) / xn → 5/2.

A: As per our company guideline we are supposed to answer only first three qs, kindly post remaining…

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: Prove that the roots of ax+4=2x²-bx are real for all values of a and b

A: The nature of roots of a quadratic equation Ax2+Bx+C=0 is:

Q: 1. Prove that fraction where m is z-4

A: Given: zn-1z-1 ⇒zn-1nz-1 ----(1) Let apply the limit in equation (1) limz→1zn-1nz-1 ----(2) Let n=pq…

Q: 15. What are all the possible rational roots of f(x) = 6x* – x² – 4x² – x – 2? ±1. ±2. ±3, ±6, ±늘,…

A: Rational root problem......

Q: Show that 1 is the only solution of x25 - 1 = 0 in Z37.

Q: Assume that (8) ∞∑n=1nanxn−1=∞∑n=0anxn∑n=1∞nanxn−1=∑n=0∞anxn for all x, and determine what this…

Q: Q:If a.ß are roots of ax +2bx +c = 0 and a + 8.ß + 8 be those of b2 - ac в— АС a Ax +2Bx + C = 0,…

Q: #8a using Descartes's rule of sign

A: Prove in the following that if all the coefficients of an equation are positive and the equation…

Q: Generalize Theorem 1.3.9 by proving that every rational solution of a polyno- mial equation x" +…

A: Give polynomial is xn+an-1xn-1+…+a1x+a0=0, where all the coefficient's ak are integers. Let p,q are…

Q: Find the residue of 3 Z 2 z² - 1 at z→ ∞0.

A: We have to find the solution of the given problem.

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: Determine the rational roots of 2x 5 - 11x 4 + 14x 3 - 2x 2 + 12x + 9 = 0. 3/2 1/2 -9/2 -1/2 -1 -3…

Q: How many irreducible polynomials of degree 4 are there in Z/2[X]?

Q: Prove that 2n+l > (n + 2) · sin(n) for all positive integers n.

Q: Prove that the function w(z), expressing the roots of the equa- tion 3ర్ - 25w° + 60u - 2 = 0 in…

A: We consider z as a parameter and for each complex value of z will look for all complex roots w of…

Q: of both P(x) and P(x)÷(x-r) P(x)= x4+2x2+8x+5; r=-1

A: if (x-r) is a factor of p(x) and p(x)÷(x-r) then when we divide both by (x-r) the remainder will be…

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

Q: Show that, if gcd(a,n)=gcd(a-1,n)=1 then a is a solution of the polynomial parity x² + ... + xó(n)-1…

Q: For a Cauchy-Euler DE the roots of auxiliary are m = -3 + 3i then Ye = x= (c, cos(3 In x)+ c2 sin(3…

Q: ot of P(x)=0 by verifying that x-r is a factor of both P(x) and P(x)/(x-r) P(x)= x^4+2x^2+8x+5;…

Q: Determine the number a in Z5 so that (x - a) becomes a divisor of f (x) = x ^ 3 + 2x ^ 2 + 3x + 3 in…

Q: 9 Chapter 3.4, Problem 187E Express the rational function as a sum or difference of two simpler…

A: Concept -factorized the rational express

Question

Show that if yª – ay
number and a # 2.
= 1 has no rational solution for positive a such that a is a natural

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

SEE SOLUTION Check out a sample Q&A here

Step 1

VIEW

Step 2

VIEW

Step by step

Solved in 2 steps

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

$Ring$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.

Similar questions

Show that the ideal is a maximal ideal of .
Let x and y be in Z, not both zero, then x2+y2Z+.
25. Prove that if and are integers and, then either or. (Hint: If, then either or, and similarly for. Consider for the various causes.)
Let be as described in the proof of Theorem. Give a specific example of a positive element of .
23. Find k such that x2+(k+1)x+k+4=0 will have two equal roots.
13. Prove that if and are rational numbers such that then there exists a rational number such that . (This means that between any two distinct rational numbers there is another rational number.)