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$$\left\{ \sum_{n=1}^{k}(a_{n}x^{b_n})+a_{k+1} \mid a_{n}\in \mathbb{R},k, b_n \in \mathbb{N}\: \mathbf{and}\: b_n \geqslant 2 \right\}$$

This is really supposed to be a vector space, but ignoring any aspect of that, is this an accurate description of the set of all polynomials with degree greater or equal to two? Or am I just very wrong? Can I accurately describe what set the numbers

$$a_1,a_2,a_3,a_4...a_n$$

lie in with just saying that $$a_n \in \mathbb{R}$$

or is there a better syntax to use?

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  • $\begingroup$ The degrees here need not be greater than 2, only the exponents. No restriction is placed on the coefficients, which is good since one needs them all zero to obtain the zero element of the vector space. These polynomials may be more simply described as those with $0$ for the linear coefficient. $\endgroup$ – coffeemath Oct 27 '15 at 21:03
  • $\begingroup$ I think your description is not right. Consider, for instance the polynomial such that $a_n = 0$ for $n=1\ldots k$ and $a_{k+1} \neq 0$. This is in fact a constant and fits the description of your set, but this polynomial has degree $0$. $\endgroup$ – Integral Oct 27 '15 at 21:08
  • $\begingroup$ The set of polynomials (in one variable) over $\mathbb R$ of degree at least two is just $$ \left\{ \sum_{n=0}^k a_nx^n : k\geq 2,a_k\neq0 \right\}$$ right? And it is NOT a vector space because it doesn't contain $0$. $\endgroup$ – MPW Oct 27 '15 at 21:10
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Your description seems too complicated and slightly incorrect.

Here is a simpler description: $$ \big\{ \sum_{k=0}^n a_k x^k : a_k \in \mathbb R, n \in \mathbb N, n>2, a_n \ne 0 \big\} $$

Note that a polynomial with degree higher than $2$ can have terms of lower degree. The key point is that the term with highest degree has exponent greater than $2$.

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  • $\begingroup$ Oh, thank you! This makes a lot more sense. I was confusing myself with my own description. $\endgroup$ – user3210986 Oct 27 '15 at 22:21
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That set doesn't contain any polynomial $P(x)$ which has a non-zero coefficient next to $x^1$. So it obviously doesn't contain all polynomials with $deg\ge 2$.

I think $a_n \in \mathbb{R}$ is ok, but I'm not a very experienced mathematician :)

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