11.6.3 Deduce from Exercise 11.6.2 a formula for "Pythagorean triples" of poly- nomials, like Euclid's formula for ordinary Pythagorean triples. It is now possible to imitate Fermat's proof, showing that r(u)* - s(u)* = v(u)² is impossible for polynomials, and hence that y? = 1–x* has no parameterization by rational functions. It follows that the same is true of certain cubic curves. REFERENCE 11.6.2 11.6.2 Convince yourself that "lines" and “slope" make sense in the rational func- tion plane, and hence show that each point # (0, –1) on the "unit circle" х(и)? + у(и)? %3D 1 is of the form 1– t(u)? x(u) 1 + t(u)?' 21(u) y(u) = 1+ t(u)? %3D for some rational function t(u). PLEASE ANSWER 11.6.3..Use 11.6.2 ..... as a REFERENCE.
11.6.3 Deduce from Exercise 11.6.2 a formula for "Pythagorean triples" of poly- nomials, like Euclid's formula for ordinary Pythagorean triples. It is now possible to imitate Fermat's proof, showing that r(u)* - s(u)* = v(u)² is impossible for polynomials, and hence that y? = 1–x* has no parameterization by rational functions. It follows that the same is true of certain cubic curves. REFERENCE 11.6.2 11.6.2 Convince yourself that "lines" and “slope" make sense in the rational func- tion plane, and hence show that each point # (0, –1) on the "unit circle" х(и)? + у(и)? %3D 1 is of the form 1– t(u)? x(u) 1 + t(u)?' 21(u) y(u) = 1+ t(u)? %3D for some rational function t(u). PLEASE ANSWER 11.6.3..Use 11.6.2 ..... as a REFERENCE.
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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Transcribed Image Text:11.6.3 Deduce from Exercise 11.6.2 a formula for “Pythagorean triples" of poly-
nomials, like Euclid's formula for ordinary Pythagorean triples.
It is now possible to imitate Fermat's proof, showing that r(u)* – s(u)* = v(u)²
is impossible for polynomials, and hence that y? = 1 – x has no parameterization
by rational functions. It follows that the same is true of certain cubic curves.
REFERENCE 11.6.2
11.6.2 Convince yourself that "lines" and “slope" make sense in the rational func-
tion plane, and hence show that each point # (0, –1) on the “unit circle"
x(u)? + y(u)? = 1
is of the form
1- t(u)?
x(u) =
1+ t(u)2'
2r(и)
У (и) %3D
1+ t(u)?
for some rational function t(u).
PLEASE ANSWER 11.6.3..Use 11.6.2
as a REFERENCE.
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