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---
id: 5900f5231000cf542c510034
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title: 'Problem 438: Integer part of polynomial equation''s solutions'
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challengeType: 5
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forumTopicId: 302109
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dashedName: problem-438-integer-part-of-polynomial-equations-solutions
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---
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# --description--
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For an $n$-tuple of integers $t = (a_1, \ldots, a_n)$, let $(x_1, \ldots, x_n)$ be the solutions of the polynomial equation $x^n + a_1x^{n - 1} + a_2x^{n - 2} + \ldots + a_{n - 1}x + a_n = 0$.
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Consider the following two conditions:
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- $x_1, \ldots, x_n$ are all real.
- If $x_1, ..., x_n$ are sorted, $⌊x_i⌋ = i$ for $1 ≤ i ≤ n$. ($⌊·⌋:$ floor function.)
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In the case of $n = 4$, there are 12 $n$-tuples of integers which satisfy both conditions.
We define $S(t)$ as the sum of the absolute values of the integers in $t$.
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For $n = 4$ we can verify that $\sum S(t) = 2087$ for all $n$-tuples $t$ which satisfy both conditions.
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Find $\sum S(t)$ for $n = 7$.
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# --hints--
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`polynomialIntegerPart()` should return `2046409616809` .
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```js
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assert.strictEqual(polynomialIntegerPart(), 2046409616809);
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```
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# --seed--
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## --seed-contents--
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```js
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function polynomialIntegerPart() {
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return true;
}
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polynomialIntegerPart();
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```
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# --solutions--
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```js
// solution required
```