Question

Question #a407d

Answer 1

`x^2014+x^2013=-x^2015=-x^2012`

Explanation:

Take the expression `x^2014+x^2013` and factor `x^2012` from each term.

`x^2014+x^2013=x^2012(x^2+x)`

Note that from the equation `x^2+x+1=0` we can state that `x^2+x=-1`. Thus,

`x^2012(x^2+x)=x^2012(-1)=-x^2012`

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Alternatively, start by factoring `x^2013` from both terms.

`x^2014+x^2013=x^2013(x+1)`

From `x^2+x+1=0` we see that `x+1=-x^2`, so:

`x^2013(x+1)=x^2013(-x^2)=-x^2015`

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Another method:

`x^2014+x^2013=x^2014+x^2013+x^2012-x^2012`

Factor `x^2012` from the first three terms:

`x^2014+x^2013+x^2012-x^2012=x^2012(x^2+x+1)-x^2012`

Since `x^2+x+1=0`, this equals

`x^2012(x^2+x+1)-x^2012=0-x^2012=-x^2012`

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Similar to this, we can add and subtract `x^2015`:

`x^2014+x^2013=x^2015+x^2014+x^2013-x^2015`

Factor `x^2013` from the first three terms:

`x^2015+x^2014+x^2013-x^2015=x^2013(x^2+x+1)-x^2015`

Using the same logic as before,

`x^2013(x^2+x+1)-x^2015=-x^2015`

Answer 2

`x^2014+x^2013 = x+1 = (1+- isqrt3)/2`

Explanation:

`x^2+x+1` is the quadratic factor of `x^3-1`.

So `x^2+x+1 = 0` implies that `x^3-1=0`.

(If necessary for clarity, multiply both sides by `x-1`.)

So we see that `x^3=1`.

`2013/3=671`, so

`x^2013 = (x^3)^671 = 1^671 = 1`

Method 1

`x^2014 = x*x^2013 = x*1=x`

`x^2014 + x^2013 = x+1`

Method 2

`x^2014+x^2013 = x^2013(x+1) = 1*(x+1) = x+1`

If you want a numerical answer , solve `x^2+x+1=0` for `x = (-1+-isqrt3)/2`

We conclude with

`x+1 = (1+- isqrt3)/2`.