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I have a math homework problem that goes like this:

Determine if the given set is a subspace of $\mathbb{P}_n$ for the appropriate value of $n$:

All polynomials of the form $p(t) = a + t^2$, where $a$ is in $\mathbb{R}$

I don't really understand what the problem means by "All polynomials of the form ...". Does that mean that if I could multiply $t^2$ and $a$ by any constants and get a "polynomial of the form"?

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  • $\begingroup$ $\mathbb P_n$ and $\mathbb R_n$ do not appear to be standard notation. Do you know what they mean in this context? $\endgroup$ Oct 7 '11 at 1:03
  • $\begingroup$ $\mathbb{P}_n$ represents the set of all polynomials of degree $n$, I believe. And $\mathbb{R}_n$ should have been $\mathbb{R}$. Sorry about that. $\endgroup$ Oct 7 '11 at 1:06
  • $\begingroup$ Some of the polynomials that qualify as being of the right form are $2+t^2$, $\frac{3}{4}+t^2$, $\pi+t^2$. Polynomials that don't qualify are for example $1+t+t^2$, $5+2t^2$. $\endgroup$ Oct 7 '11 at 1:11
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You're parsing the sentence wrong. It does not refer to a $$ \text{polynomial of the form}. $$ It refers to a $$ \text{polynomial of the }{\Big(}\text{form }p(t) = a+t^2{\Big)}. $$ I.e. a polynomial that has that particular form. The particular form is that it's a constant plus $t^2$.

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  • $\begingroup$ so the set of polynomials that satisfy this are are basically just the set of any $\mathbb{R}$ number plugged in for $a$, added to $t^2$? $\endgroup$ Oct 7 '11 at 1:09
  • $\begingroup$ Yes. $\;\;\;\;\;\;$ $\endgroup$ Oct 7 '11 at 1:12
  • $\begingroup$ OK, Sorry for asking such an elementary question. $\endgroup$ Oct 7 '11 at 1:15

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