a.) Show that the set $W$ of all polynomials in $P_2$ such that $p(1)=0$ is a
subspace of $P_2$.
b.) Make a conjecture about the dimension of $W$.
c.) Confirm your conjecture by finding a basis for $W$.
I know how to show $W$ is a subspace of $P_2$, by showing closure under addition and multiplication by a scalar. However, I am clueless as to how to find a basis. I can't see how this is sufficient information to answer the question. All I can see is that if $p(1)=0$, then $a_01+a_1x+a_2x^2=0$ implies $$a_0+a_1+a_2=0$$
Best Answer
A polynomial $p(x)=ax^2+bx+c$ satisfies $p(1)=0$ if and only if $$ a+b+c=0 $$ Hence every $p\in W$ is of the form $$p(x)=ax^2+bx+(-a-b)=a(x^2-1)+b(x-1)\tag{1}$$ Can you use equation $(1)$ to find a basis for $W$ and thus compute $\dim W$?
Sidenote: There is a more fun way to do this problem. The map $T:P_2\to \Bbb R$ given by $T(p)=p(1)$ is linear and $\ker T=W$. Since $T(1)=1$ it is clear that $T$ has rank $1$. The Rank-Nullity theorem then gives $\dim W$.