What is sum of roots of polynomial f=x^3-x-1?

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You may detemine the sum of roots of a polynomial in two ways, either you may calculate the roots, or you may use Vieta's formulas.

Selecting the second method mentioned (Vieta's formulas), you need first identify how many roots the polynomial does have. Since the maximum power of unknown is 3, the polynomial has three roots, `x_(1,2,3).`

You need to remember the Vieta's formula for a polynomial of 3rd oredr,`P = ax^3 + bx^2 + cx + d` , such that:

`x_1 + x_2 + x_3 = -b/a`

`x_1x_2 + x_1x_3 + x_2x_3 = c/a`

`x_1x_2x_3 = -d/a`

The problem provides the polynomial `f = x^3-x-1` , hence, you may identify the coefficients `a,b,c,d` , such that:

`a = 1, b = 0, c= -1, d = -1`

The problem requres for you to find the sum of roots, hence, you need to use the first Vieta's relation, such that:

`x_1 + x_2 + x_3 = -b/a `

`x_1 + x_2 + x_3 = -0/1`

`x_1 + x_2 + x_3 = 0`

Hence, evaluating the sum of the roots of the given polynomial, using Vieta's formulas yields `x_1 + x_2 + x_3 = 0.`