Expressing polynomial equations as a set of linear equations

Question

Suppose the $4$-vector $c$ gives the coefficients of a cubic polynomial $p(x) = c_1 + c_2x+c_3c^2+c_4c^3$. Express the conditions

$p(1) = p(2), p'(1) = p'(2)$

as a set of linear equations of the form $Ac=b$. Give the sizes of A and b, as well as their entries.

So far, what I've done was sub in the values and differentiate the polynomials and let it equal to each other after subbing their values.

I get

$c_1 + c_2 + c_3 + c_4 = c_1 + 2c_2 + 4c_3 + 8c_4$

then,

$0 = c_2 + 3c_3 + 7c_4$

as well as

$c_2+2c_3+3c_4 = c_2 + 4c_3+12c_4$

then,

$0 = 2c_3 + 9c_4$

I am a bit lost oh what to do now.

Answer 1

So you have \begin{align} 0c_1 + 1c_2 + 3c_3 + 7c_4 &= 0\\ 0c_1 + 0c_2 + 2c_3 + 9c_4 &= 0 \end{align}

Now write these two equations of four variables in matrix-vector form. You already know that $c$ is $4\times 1$, that is, $4$ rows and $1$ column. What should the sizes of $A$ and $b$ be?