# What is meant by "All Polynomials of the form $p(t) = a + t^2$?

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"?

• $\mathbb P_n$ and $\mathbb R_n$ do not appear to be standard notation. Do you know what they mean in this context? Oct 7 '11 at 1:03
• $\mathbb{P}_n$ represents the set of all polynomials of degree $n$, I believe. And $\mathbb{R}_n$ should have been $\mathbb{R}$. Sorry about that. Oct 7 '11 at 1:06
• 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 $1+t+t^2$, $5+2t^2$. Oct 7 '11 at 1:11

## 1 Answer

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 $t^2$.

• 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 $t^2$? Oct 7 '11 at 1:09
• Yes. $\;\;\;\;\;\;$ Oct 7 '11 at 1:12
• OK, Sorry for asking such an elementary question. Oct 7 '11 at 1:15