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I saw a mathematical problem where was given a wrong definition and was asked if some numbers satisfies the definition. Is this solution fine if there is a mistake in the definition?

A real number is algebraic if it is a root of some polynomial with integer coefficients. These polynomials are of the form $$P(x)=a_0+a_1x+a_2x^2+\cdots + a_nx^n$$ where the degree of the polynomial $n=1,2,3,\ldots$ and the coefficients $a_0,a_1,a_2,\ldots,a_n$ are integers.

Prove that the numbers $x=2/3$, $x=\sqrt 3$, $x=2+\sqrt{3}$, and $x=\sqrt{2}+\sqrt{3}$ are algebraic.

My solution is just to take $P(x)=0$.

Now the problem seems to be that zero polynomial satisfies the first sentence of the definition but not the second one. So what should a student answer as the first sentence of the definition and its explanation does not corresponds to each others?

This question was from the Finnish matriculation examination on autumn 2016.

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    $\begingroup$ Two points. First, I wonder if this belongs on math.SE. Second, your BEST bet is probably to answer the question as obviously intended, and then remark at the very end about your other answer. (I say obvious, because if you didn't know what was intended, you wouldn't possibly have given the solution you give.) $\endgroup$
    – kcrisman
    Dec 22 '16 at 14:46
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    $\begingroup$ I do not quite understand what your question is. To me, the issue with this problem's phrasing is that it does not say what "degree" actually means; yes, it corresponds to the letter $n$, but the phrasing here -- if not previously encountered by a student? -- makes me wonder whether they could consider $P(x) = 0 + 0 \cdot x^1$. In this case, $n = 1$ (constraint satisfied) and both $a_0$ and $a_1$ are integers (constraint satisfied). So, I guess this is a polynomial of the desired sort (?) and $P(x) = 0$. [Again: Not clear what you are asking, though.] $\endgroup$
    – Benjamin Dickman
    Dec 23 '16 at 0:49
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    $\begingroup$ The zero polynomial doesn't satisfy $n=1,2,3,\cdots$. See my answer for more details. $\endgroup$
    – user6648
    Dec 23 '16 at 12:56
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    $\begingroup$ I think we can assume "degree" is meant to be known (as in, if a student doesn't know, then, tough, it's an entrance exam and the university expects you to). Likewise for the other terms except algebraic, which is being defined. In that case, the first sentence does not stand alone, since the class of polynomials with integer coefficients is being restricted in the second sentence. But I think it's pretty clear what a student should do in such a case: give an answer that meets all the conditions, even if there are clearer ways to state the problem. $\endgroup$
    – Raciquel
    Dec 28 '16 at 16:34
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    $\begingroup$ The original question sheet can be found here in Finnish. The translation in the question is good, the same ambiguity is present in the Finnish original. // As someone who has studied and taught in a Finnish high school, I strongly believe that you are supposed to know what the degree of a polynomial is. $\endgroup$
    – Joonas Ilmavirta
    Dec 29 '16 at 19:11
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Yeah it looks like technically the zero polynomial satisfies the constraints of the problem (see Benjamin Dickman's comment), but just saying to take $P(x) = 0$ misses the point of the examination. You are being assessed, and listing just the zero polynomial doesn't demonstrate that you know what the examiner is testing you for. The point of this question, or any question on any test for that matter, is just to give you a prompt to demonstrate that you know what you are doing. I would imagine that a good response that both demonstrates your aptitude and acknowledges what you pointed out could be worded like,

By the definition of an algebraic number given, each of these numbers is algebraic because they are each a root of the zero polynomial. But each is also the root of a non-zero polynomial. ...

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The zero polynomial doesn't satisfy the degree constraint because the degree of a polynomial is the highest power of $x$ with a nonzero coefficient. If a student doesn't know this, it doesn't make $P(x) \equiv 0$ a valid answer.

The problem you post says:

...where the degree of the polynomial $n=1,2,3,\dots$

To me that sounds like a (bad) way of saying this, and maybe our interpretations of this is where we differ:

...where the degree of the polynomial is $n$ for some integer $n \ge 1$.

The zero polynomial doesn't satisfy this constraint. The zero polynomial is considered to have undefined degree, or degree $-1$, or degree $-\infty$, depending on context. So the zero polynomial just isn't a legitimate answer to the problem, but I can't tell if this is what you're actually asking.

My solution is just to take $P(x)=0$. Now the problem seems to be that zero polynomial satisfies the first sentence of the definition but not the second one.

Then why would you give the zero polynomial as a solution when you know it doesn't satisfy the problem requirements?

So what should a student answer as the first sentence of the definition and its explanation does not corresponds to each others?

I'm not sure I understand what that means. If a student's answer doesn't satisfy all the constraints, then it's not a correct answer.

Finally, what is the mistake that you mentioned a couple times?

Is this solution fine if there is a mistake in the definition?

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This is not a wise way for a student to respond to a test question. It seems like you know that the standard definition of a polynomial of $n$th degree requires that $a_n \ne 0$ (even though you didn't explicate that observation in your question).

So: You know what the standard definition is. You can picture what is probably on the answer sheet for the test (which will not be the zero function). Relying on a curt, technical "gotcha" due to a typo is neither going to display how much you know, nor synchronize with the real definition, nor match the official answer sheet. There are occasional cases where students have challenged official answers after the fact and succeeded, but those are very, very rare (that being an enormous judicial challenge).

Immensely better if you: (a) point out that you know what the standard definition is, and (b) write a more sophisticated answer compliant with the the standard definition, and (c) make it easier for the graders to match up with the predefined grading rubric.

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  • $\begingroup$ Yes. But I have seen sometimes different mathematical competition problems where were given different definitions for example what is a good number. In these problems I had to use the definition given in the problem instead of standard definition of a good number, if there is that definition at all. $\endgroup$
    – just seen
    Dec 31 '16 at 7:59

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