Find a basis and dimension for the vector space $V=\bigl\{f(x)\in\mathbb{R}[x];\ \deg f<5,\ f''(0)=f(1)=f(-1)=0\bigr\}.$ $$V=\bigl\{f(x)\in\mathbb{R}[x];\ \deg f<5,\ f''(0)=f(1)=f(-1)=0\bigr\}.$$
NOTE:
The degree is strictly less than 5 not equal.
Also my prof and T.A. said I should do $f(1)=f(-1)$ then solve it then do $f''(0)=0$ somewhere along the lines but I don't understand how to go about doing it that way.
PLEASE!! Can someone solve it doing it that way and not using matrices because for these questions you don't need to use matrices to solve it.
 A: Hint: 
Write $$
f(x)=a_0+a_1x+\cdots+a_4x^4.
$$
Find the relations among the $a_i$'s using $$f''(0)=f(1)=f(-1)=0.$$ For instance, what can you tell by $f(1)=0$? 
Once you have the relations (precisely, linear equations) for these $a_i$'s, you would get a subspace of $\mathbb{R}^5$, the dimension of which is what you are looking for. 
A: Because $f(1)=0,$ you know $x-1$ is a factor of $f(x).$ Because $f(-1)=0,$ you know $x+1$ is a factor of $f(x).$ Therefore $(x-1)(x+1)=x^2-1$ is a factor of $f(x).$
Knowing that $f(x)$ is a polynomial of degree at most $4,$ and has a factor of $x^2-1,$ we must have $f(x)=(x^2-1)(ax^2+bx+c)$ for some constants $a,$ $b,$ and $c.$ 
Now consider the second derivative of $f(x)$; it is a polynomial of degree two or less, something of the form $g(x)=px^2+qx+r,$ and the value of $r$ is a multiple of the coefficient of $x^2$ in $f(x).$ But we know $g(0)=0,$ so $r=0.$ Therefore the coefficient of $x^2$ in $f(x)$ is zero.
Work out the coefficient of $x^2$ in $f(x)$ in terms of $a,$ $b,$ and $c.$ (Hint: $b$ isn't involved.) Express the fact that this coefficient is zero. The result will be an equation that lets you eliminate either $a$ or $c.$
To recapitulate: $f(1)=f(-1)=0$ lets you factor $f(x),$ leaving only three unknown parameters, and $f''(0)=0$ gives you a relationship among those unknowns, allowing you to reduce them to two.
I think writing $f(x)$ in its factored form makes it especially simple to rewrite it as a linear combination of two polynomials.
