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Find matrix and Jordan basis of endomorphism $f \in L(\mathbb{R}[x]_3)$ for which $ker\ f = span(1,x)$, $f\circ f=f$, $f(x^2) = 1-x+x^2$, $f(x^3)=p$ and $p(1)=p'(1)=0$. From $ker\ f = span(1,x)$ and $f(x^2) = 1-x+x^2$ we know that matrix of f is $M = \begin{bmatrix} 0 & 0 & 1 & a_{14} \\ 0 & 0 & -1 & a_{24} \\ 0 & 0 & 1 & a_{34} \\ 0 & 0 & 0 & a_{44} \end{bmatrix}$. Let $p(x) = a + bx + cx^2 +dx^3$. From $p(1)=p'(1)=0$ we have $a + b + c +d =0$ and $b + 2c +3d= 0$, so $p(x) =a + bx + (-3a-2b)x^2+(2a+b)x^3$. How to finish it?

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you really ought to insert your coefficients for $p(x)$ in the correct places in $M,$ and you have ignored $$ M^2 = M. $$

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