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.