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Question

Given a polynomial $$ f(x) = \prod_{i=1}^n (x-\lambda_i) = \sum_{i=0}^n a_i \, x^i $$ with known roots $\lambda_i$ and coefficients $a_i$ and consider the truncated polynomial $$ \tilde{f}(x) = \sum_{i=0}^m a_i \, x^i $$ with $0<m<n$. Is there a connection between the roots of $f$ and $\tilde{f}$?

Example

The polynomial $$x^4+12x^3+49x^2+78x+40$$ has the roots $\lambda_i \in \{-1,-2,-4,-5\}$. Does this information help in any way to calculate the roots of the truncated polynomial $$12x^3+49x^2+78x+40$$ ?

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    $\begingroup$ Unfortunately, no. $\endgroup$ Commented Sep 8, 2020 at 12:21

2 Answers 2

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Your example already shows that there is no obvious connection. The roots of $x^4+12x^3+49x^2+78x+40$ are all integers, whereas the truncated polynomial $12x^3+49x^2+78x+40$ has no rational root and only one real root.

Truncating a polynomial drastically changes several properties of the polynomial.

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$$bx+c=0\iff x=-\frac bc,$$

$$ax^2+bx+c=0\iff x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}.$$

You don't see any easy relation. Adding terms adds degrees of freedom and the roots can wildly vary.

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