# Vector space of polynomials

Do all polynomials $ax^3 + bx^2 + cx + d$ with a root at $x=1$ form a vector space? Do the coefficients $(a,b,c,d)$ form a vector space?

My reasoning: Since $x=1$ is a root, we can't have $(a,b,c,d)$ all zeros. The space doesn't include the zero-matrix. Hence, it's not a vector space

Suppose, $f(x)$ and $g(x)$ = polynomials of form $ax^3 + bx^2 + cx + d$ with root at $x=1$ $h(x) = f(x) + g(x)$ Since $x=1$ is a root, $f(1) = 0$ and $g(1) = 0$. $h(1) = f(1) + g(1) = 0$ ($h(x)$ also has a root at $x = 1$)

I'm not sure about the zero matrix in this case. What is considered as "zero matrix" in polynomials?

Thanks for the help in advance.

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No. If you mean by "zero matrix" the zero vector, I don't understand why you think the set of polynomials with root at $1$ does not contain the zero vector. The zero vector $(0,0,0,0)$ corresponds to the zero polynomial and is certainly also $0$ at $1$. –  Matt N. Sep 19 '12 at 12:04

Hint: For a polynomial $ax^3+bx^2+cx+d$ having a root at $x=1$ is equivalent to have $a+b+c+d=0$.
The zero vector in this case is $(0,0,0,0)$. This corresponds to the polynomials $p(x) = 0$. You have $p(1) = 0$ hence $p$ is in the set of all polynomials that are zero at $1$. So your set does contain the zero vector.