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This is a very elementary question, but one I haven't happened to have crossed for a while I guess. I'm helping my niece out with some basic math and we were going through polynomials in her math book. In my head, I always see polynomial as expression involving terms such as $ax^ny^m$.

By wikipedia:

A polynomial expression is an expression that can be built from constants and symbols called variables or indeterminates by means of addition, multiplication and exponentiation to a non-negative integer power.

Now my niece has a task, where she needs to identify whether an expression is, or is not a polynomial and she has two examples:

$$x+y-z\;\;\;\;(1)$$ and $$x+y+z\;\;\;\;(2)$$

Now the math book says that $(1)$ is not a polynomial, but $(2)$ is and this was a bit confusing to myself as well. So is the only reason why $(1)$ is not a polynomial simply because $(1)$ has the $-1$ multiplier in front of variable $z$ (or they also could be constants?) so it's subtraction and not addition as in the definition? Is it that simple? Or is there something wrong with the book?

UPDATE: I added the screenshots from the book with translations. In the book they use symbols $a,b,c$ in place of $x,y,z$ I used in the question.

enter image description here

jjepsuomi
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    The book is wrong because they're both polynomials. – CyclotomicField Nov 16 '22 at 20:08
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    Haha, thank you, I was going like "what have I been doing all these years". – jjepsuomi Nov 16 '22 at 20:08
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    Of course (1) is a polynomial. Anyone who says otherwise is incompetent. – KCd Nov 16 '22 at 20:09
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    What book (or reference) are you using? That seems like a weird error. – lulu Nov 16 '22 at 20:14
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    @lulu Thank you all. The book is a national secondary school math book. I think this was an isolated error in the book. I will give back the feedback :) – jjepsuomi Nov 16 '22 at 20:18
  • might you do maths educators SE? maybe they can shed some insight why some textbooks might not consider it a polynomial like what Lee Mosher pointed out. or maybe what the book means is that it's not in 'polynomial form' or something but the way Lee Mosher writes it is in 'polynomial form'. Edit: what's the definition of exactly of polynomial in that textbook? – BCLC Nov 16 '22 at 20:19
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    Could you supply the reference? Typos and such are inevitable, and usually not a big deal, but this seems more egregious. – lulu Nov 16 '22 at 20:19
  • @lulu Hi, I found the book from here (a Finnish text book for secondary school students): https://www.prisma.fi/fi/prisma/hassinen-kuutio-x-oma-ops-2016-pehmeakantinen-kirja

    Sorry that it's in Finnish, I don't know if you can find electronic version of this. I can provide a picture tomorrow of the actual definitions and the task itself (have to add translation as well)

     – jjepsuomi Nov 16 '22 at 20:26
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    @BCLC Thank you for the help, I can try this out tomorrow. I need to check the definition tomorrow, but if I recall it correctly (translated from Finnish) the definition was: "Polynomial is a mathematical expression containing constants and variables, where the variables can't contain non-positive and non-integer powers". – jjepsuomi Nov 16 '22 at 20:30
  • jjepsuomi i don't see how both aren't polynomials. probably a mistake esp if it's just yes/no or true/false. – BCLC Nov 21 '22 at 01:06
  • @BCLC I added screenshot of the task – jjepsuomi Nov 23 '22 at 11:02

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You can point out to your niece that $$x + y - z = x + y + (-1) z $$ and now it fits the definition perfectly, with $-1$ being a constant.

Lee Mosher
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