The space $\displaystyle {P}_{{2}}$ represents all 2nd degree polynomials. Recall that a polynomial such as $\displaystyle {p}{\left({x}\right)}={3}+{1}{x}+{4}{x}^{{2}}$ would be the vector $\displaystyle {\left[\begin{array}{c} {3}\\{1}\\{4}\end{array}\right]}$ in $\displaystyle {P}_{{2}}$. The standard basis polynomials for this space are $\displaystyle {\left\lbrace{1},{x},{x}^{{2}}\right\rbrace}$.

The function $\displaystyle {F}$, defined by $\displaystyle {F}{\left({p}{\left({x}\right)}\right)}={\left({x}+{8}\right)}\cdot{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{p}{\left({x}\right)}$, is a linear transformation from $\displaystyle {P}_{{2}}$ to $\displaystyle {P}_{{2}}$. It takes the derivative of $\displaystyle {p}{\left({x}\right)}$ and then multiplies the result by $\displaystyle {\left({x}+{8}\right)}$.

1. Write the matrix for this linear transformation according to the standard basis polynomials. [Hint: Find where the standard basis polynomials go under this transformation.]
   
   
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2. Write the polynomial $\displaystyle {2}+{7}{x}+{8}{x}^{{2}}$ as a vector.
   
   
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3. Then, use your matrix from above to calculate $\displaystyle {F}{\left({2}+{7}{x}+{8}{x}^{{2}}\right)}$. Write your answer as a polynomial.
   
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