Negate the Roots

Algebra Level 3

The roots of the monic polynomial x5+ax4+bx3+cx2+dx+e x^5 + a x^4 + b x^3 + c x^2 + d x + e are r1 -r_1, r2-r_2, r3-r_3, r4-r_4, and r5-r_5, where r1r_1, r2r_2, r3r_3, r4r_4, and r5r_5 are the roots of the polynomial x5+9x4+13x357x286x+120.x^5 + 9x^4 + 13x^3 - 57 x^2 - 86 x + 120. Find a+b+c+d+e. |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 2x12 2x - 12 .

A polynomial is monic if its leading coefficient is 1. For example, the polynomial x3+3x5 x^3 + 3x - 5 is monic but the polynomial x4+2x36 -x^4 + 2x^3 - 6 is not.

The notation | \cdot | denotes the absolute value. The function is given by x={xx0xx<0 |x | = \begin{cases} x & x \geq 0 \\ -x & x < 0 \\ \end{cases} For example, 3=3,2=2 |3| = 3, |-2| = 2 .

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