selecting steps that are not valid (that is, they do not follow from the assumptions or from earlier steps). Proof: Assume that æ is an integer that satisfies æ(x* + æ² – æ + 1) = -2. (1) Rearranging this equation, we obtain æ³ + æ³ – a² + æ + 2 = 0. (2) The polynomial in this equation can be factored as follows: æ³ + æ³ (2³ – a? + 1)(æ² +x +2). (3) This implies that a – a² + 1 = 0 and a? + æ +2 = 0. 2² + æ + 2 = (4) The second equation in step (3) is a quadratic equation with negative discriminant. (5) This implies that no real number x satisfies the second equation in step (3). (6) Therefore equation x(x4 + x² – x +1) = -2 does not have any solutions in real numbers. Select the step or steps in this argument that are invalid: (1) O (2) (3) (4) (5) (6)
selecting steps that are not valid (that is, they do not follow from the assumptions or from earlier steps). Proof: Assume that æ is an integer that satisfies æ(x* + æ² – æ + 1) = -2. (1) Rearranging this equation, we obtain æ³ + æ³ – a² + æ + 2 = 0. (2) The polynomial in this equation can be factored as follows: æ³ + æ³ (2³ – a? + 1)(æ² +x +2). (3) This implies that a – a² + 1 = 0 and a? + æ +2 = 0. 2² + æ + 2 = (4) The second equation in step (3) is a quadratic equation with negative discriminant. (5) This implies that no real number x satisfies the second equation in step (3). (6) Therefore equation x(x4 + x² – x +1) = -2 does not have any solutions in real numbers. Select the step or steps in this argument that are invalid: (1) O (2) (3) (4) (5) (6)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
Transcribed Image Text:Below you are given an attempted proof of the following statement: "Equation x (x² + x².
x + 1) = -2 does not have any solutions in real numbers". Find all errors in this proof by
selecting steps that are not valid (that is, they do not follow from the assumptions or from earlier
steps).
Proof: Assume that x is an integer that satisfies x(x* + ² – x + 1) = -2.
%3D
(1) Rearranging this equation, we obtain æ³ + æ³
(2) The polynomial in this equation can be factored as follows: a + x³ – x2 + x + 2 =
(2³ – a2 + 1)(x² +x +2).
(3) This implies that æ – a? +1 = 0 and x? + + 2 = 0.
x2 + x + 2 = 0.
-
(4) The second equation in step (3) is a quadratic equation with negative discriminant.
(5) This implies that no real number æ satisfies the second equation in step (3).
(6) Therefore equation x (x4 + x? – x +1) = -2 does not have any solutions in real
numbers.
Select the step or steps in this argument that are invalid:
(1)
(2)
(3)
(4)
O (6)
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