# 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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