Let $P^3_1$ be a set of polynomial over real number , $\mathbb{R}$ , of degree $3$ or less which has $1$ as a solution together with zero polynomial ; namely,
$P^3_1=\left\{\,p(x)=a+bx+cx^2+dx^3|a,b,c,d\in \mathbb{R} ;p(1)=0\right\} \cup \left\{\,0\right\} $
Define operation $+$ and $\cdot$ as follow;
For $p(x),q(x)\in P^3_1$ such that $p(x)=a_1+b_1x+c_1x^2+d_1x^3$ and $q(x)=a_2+b_2x+c_2x^2+d_2x^3$ define
$p(x)+q(x)=(a_1+a_2)+(b_1+b_2)x+(c_1+c_2)x^2+(d_1+d_2)x^3$ and
$r\cdot p(x)=ra_1+rb_1x+rc_1x^2+rd_1x^3$
Is $P^3_1$ a vector space over $\mathbb{R} $ under these operation? Justify your answer.
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