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\noindent {\bf Problem statement} Polynomials have roots. For example,
$(x-1) (x+2) (x-3) = x^3 -2x^2 -5x + 6 $ which means that $x^3 -2x^2
-5x + 6 $ is $0$ when $x=1$ or $x=-2$ or $x=3$. Low degree polynomials
have algebraic recipes for roots in terms of their coefficients. There
are no such formulas for higher degree polynomials. If $r_1$, $r_2$,
and $r_3$ are the roots of a third degree polynomial, $x^3 + Ax^2 +Bx
+C$, then the coefficients ($A$, $B$, and $C$) are functions of the
roots ($r_1$, $r_2$, and $r_3$).

\medskip

\noindent a) What are the functions?  That is, write $A$, $B$, and $C$
as functions of $r_1$, $r_2$, and $r_3$. Verify if $\displaystyle
\cases {r_1 = 1&\cr r_2= -2&\cr r_3= 3&\cr}\!\!\!\!$ then
$\displaystyle \cases {A = -2&\cr B= -5&\cr C= 6&\cr}\!\!\!\!$.

\medskip

\noindent b) Suppose the roots are changed: $\displaystyle \cases
{r_1: 1 \to 1.02&\cr r_2: -2 \to -2.04 &\cr r_3: 3 \to
2.95&\cr}\!\!\!\!$.  Use partial derivatives and linearization to
predict the {\it approximate} changes in the coefficients.

\medskip

\noindent c) Suppose now that the coefficients are changed. That is,
consider new coefficients: $\displaystyle \cases {A = -2.03 &\cr B=
-5.02&\cr C= 6.01&\cr}\!\!\!\!$.  Approximate the roots which would
give these coefficients. (This is harder, and a {\it new idea}\/ is
needed: what perturbations in the roots will, to first order, give
these perturbations in the coefficients? A ``system'' of three linear
equations in three unknowns must be solved.)

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