Another way to get to the answer:
$P_3=\{{ax^3+bx^2+cx+d:a,b,c,d{\rm\ in\ }{\bf R}\}}$. For $p(x)=ax^3+bx^2+cx+d$ in $P_3$, $p(1)=0$ is $a+b+c+d=0$ So, you have a "system" of one linear equation in 4 unknowns. Presumably, you have learned how to find a basis for the vector space of all solutions to such a system, or, to put it another way, a basis for the nullspace of the matrix $\pmatrix{1&1&1&1\cr}$ One such basis is $\{{(1,-1,0,0),(1,0,-1,0),(1,0,0,-1)\}}$ which corresponds to the answer $\{{x^3-x^2,x^3-x,x^3-1\}}$ one of an infinity of correct answers to the question.