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QUESTION A quadratic polynomial $p(x)$ in $x$ is frequently
written in the form $p(x)=ax^2+bx+c$. Sometimes, however, it is
more convenient to write it in the form $$p(x)=\alpha
x^{(2)}+\beta x^{(1)}+\gamma x^{(0)}$$ where
$x^{(2)}=x(x-1),x^{(1)}=x,x^{(0)}=1.$

For example, $5x^2-8x+2=5x(x-1)-3x+2=5x^{(2)}-3x^{(1)}+2.$

Express the following polynomials in this form:

\begin{description}

\item[(a)]
$25x^2+4x-7$;

\item[(b)]
$-6x^2+14x+3$.

\item[(c)]

Show also that any quadratic polynomial with real coefficients can
be written in this form.
\end{description}

[So $\{x^{(2)},x^{(1)},x^{(0)}\}$ is a basis for the set of
polynomials of degree two or less.]

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ANSWER

\begin{description}
\item[(a)]
$25x^{(2)}+29x^{(1)}-7x^{(0)}$,

\item[(b)]
 $-6x^{(2)}+8x^{(1)}+3x^{(0)}$,

\item[(c)]
$ax^{(2)}+(a+b)x^{(1)}+cx^{(0)}$.

\end{description}


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