Negate the Roots

Algebra Level 3

The roots of the monic polynomial \[ x^5 + a x^4 + b x^3 + c x^2 + d x + e \] are \( -r_1\), \(-r_2\), \(-r_3\), \(-r_4\), and \(-r_5\), where \(r_1\), \(r_2\), \(r_3\), \(r_4\), and \(r_5\) are the roots of the polynomial \[x^5 + 9x^4 + 13x^3 - 57 x^2 - 86 x + 120.\] Find \( |a+b+c+d+e|.\)

Details and assumptions

A root of a polynomial is a number where the polynomial is zero. For example, 6 is a root of the polynomial \( 2x - 12 \).

A polynomial is monic if its leading coefficient is 1. For example, the polynomial \( x^3 + 3x - 5 \) is monic but the polynomial \( -x^4 + 2x^3 - 6 \) is not.

The notation \( | \cdot | \) denotes the absolute value. The function is given by \[ |x | = \begin{cases} x & x \geq 0 \\ -x & x < 0 \\ \end{cases} \] For example, \( |3| = 3, |-2| = 2 \).

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