Continued Fractions Approximation <mrow class="MJX-TeXAtom-ORD">

uri2e4g 2022-07-01 Answered
Continued Fractions Approximation
x 2 + 3 x + 2 x 2 x + 1
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Answers (1)

Alisa Jacobs
Answered 2022-07-02 Author has 13 answers
x 2 + 3 x + 2 x 2 x + 1 = 1 + 4 x + 1 x 2 x + 1 = 1 + 1 1 4 x 5 16 + 21 16 4 x + 1 = 1 + 1 1 4 x 5 16 + 1 64 21 x + 16 21
At each stage, we are doing a polynomial division instead of an integer division, but otherwise, the process is the same as with continued fractions with integers.
We can get the Bezout polynomials by truncating the continued fraction:
1 + 1 1 4 x 5 16 = 4 x + 11 4 x 5
That is, we can write the polynomial GCD (a constant since they are relatively prime) as
( 4 x + 11 ) ( x 2 x + 1 ) ( 4 x 5 ) ( x 2 + 3 x + 2 ) = 21
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