functionsFind the good result fof(x) = x-1+1/(x^2+x+1) fof(x) = 0

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to use the information provided by the problem, such that:

`{((fof)(x) = (x-1)+1/(x^2+x+1)),((fof)(x) = 0):} => (x-1)+1/(x^2+x+1) = 0`

You need to bring the terms in equation to a common denominator, such that:

`(x - 1)(x^2 + x + 1) + 1 = 0`

You need to use the following formula, such that:

`(x - 1)(x^2 + x + 1) = x^3 - 1`

Substituting ` x^3 - 1` for `(x - 1)(x^2 + x + 1)` yields:

`x^3 - 1 + 1 = 0 => x^3 = 0 => x = 0`

Hence, evaluating the solution to the given equation, under the given conditions, yields `x = 0.`

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

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We notice that the denominator x^2+x+1 is a factor of the product;

(x - 1)(x^2+x+1) = x^3 - 1

We'll divide by x - 1:

x^2+x+1 = (x^3 - 1)/(x - 1)

We'll substitute the denominator by the equivalent ratio:

fof(x) = x - 1 + 1/[(x^3 - 1)/(x - 1)]

fof(x) = x - 1 +  (x - 1)/[(x^3 - 1)]

We'll factorize by x - 1:

fof(x) = (x - 1)[1 + 1/(x^3 - 1)]

fof(x) = (x - 1)[(x^3 - 1 + 1)/(x^3 - 1)]

We'll eliminate like terms:

fof(x) = (x - 1)[(x^3)/(x^3 - 1)]

We'll put fof(x) = 0:

(x - 1)[(x^3)/(x^3 - 1)] = 0

We'll set each factor as zero:

x - 1 = 0

x = 1

x^3 = 0

x = 0

The solutions of the equation are: {0 ; 1}.

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