Prove the following statement using direct, contrapositive, or proof by contradiction. We say that a point P = (x,y) in R^2 is rational if both x and y are rational. More precisely, P is rational if P = (x,y) is within Q^2. An equation F(x,y) = 0 is said to have a rational point if there exists x_0,y_0 within set Q such that F(x_0,y_0) = 0. For example, the curve x^2 + y^2 - 1 = 0 has the rational point (x_0,y_0) = (1,0). Show that the curve x^2 + y^2 - 3 = 0 has no rational points.

Prove the following statement using direct, contrapositive, or proof by contradiction. We say that a point P = (x,y) in R^2 is rational if both x and y are rational. More precisely, P is rational if P = (x,y) is within Q^2. An equation F(x,y) = 0 is said to have a rational point if there exists x_0,y_0 within set Q such that F(x_0,y_0) = 0. For example, the curve x^2 + y^2 - 1 = 0 has the rational point (x_0,y_0) = (1,0). Show that the curve x^2 + y^2 - 3 = 0 has no rational points.

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 14E

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