Question on Bezout’s Lemma

For a question such as on Homework sheet 3 1ii.) and on some of the past papers, is it strictly necessary to use Bezout’s lemma? Would a simple substitution of f and g not suffice?

Also, would you get full marks for using a substitution instead of using Bezout’s Lemma.

Thanks

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Reply:

You would get full marks for any correct proof or answer clearly written.

Note that Bezout’s lemma is merely an `explicit Euclidean algorithm.’ As soon as f and g get at all large, you’ll find that the Euclidean algorithm is much more efficient than direct substitution. In some suitably general sense, the Euclidean algorithm is one of the most efficient algorithms around. It’s a bit of a miracle, in some sense.

However, your question is a good one. Check for yourself what the procedure of finding h, k by substitution would involve.

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