Continued Fractions Approximation x 2 + 3 x + 2 x 2 − x + 1

Question

Continued Fractions Approximation                              x          2                +        3        x        +        2                              x          2                −        x        +        1

Alisa Jacobs · Accepted Answer

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

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

Answered question

Answer & Explanation

New Questions in Calculus 1