Calculate (z1+1)^10+(z2+1)^10. z1, z2 are the solutions of equation z^2+z+1.

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neela | High School Teacher | (Level 3) Valedictorian

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Let the  solution of z^2+z+1= 0 be z1and z2.

To find (z1+1)^10+(z2+1)^10.

Solution:

Z1 and z2 are given to be the roots of z^2+z+1 = 0

So, (z1+1) and  (z2+1) must be the roots of the equation, (z-1)^2+(z-1)+1 = 0.

Therefore  z^2-2z+1+(z-1)+1 = 0. Simplifying , we get

 z^2-z+1 = 0 which has the roots z1+1 = u and z2+1 =  v.

Therefore  u+v = 1 and uv = 1.

Now multiply (z^2-z+1) by (z+1) and we get z^3+1.

Therefore z^2-x+1 = 0 has the same roots of (z^2-z+1)(z+1) = 0. Or z^3-1 = 0

Therefore z^3 = -1, has 3 roots:  -1 , u and v.

So u^3 = -1 and  v^3 = -1.

Therefore u^10 + v^10 = u*(u)^9+ v*(v)^9

u^10+v^10 = u (u^3)^3+v(v^3)^3

u^10+v^10 = u(-1)+ v(-1) , u^3 = v^3 = -1 , being the roots of z^3= -1.

Therefore u^10+v^10 = -(u+v) = - (1) = -1.

u^3+v^3 = (z1+1)^3+(z2+1)^3 = -1.

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

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If z^2 + z + 1 = 0, that means that if we'll multiply both sides by (z-1), we'll get:

(z-1)(z^2 + z + 1) = 0

But the product is the result of difference of cubes:

z^3 - 1 = (z-1)(z^2 + z + 1)

If (z-1)(z^2 + z + 1) = 0, then z^3 - 1 = 0

We'll add 1 both sides:

z^3 = 1

z1 and z2 are the roots of the equation z^3 - 1 = 0, so:

z1^3 = 1

z2^3  = 1

From the equation z^2 + z + 1 = 0

z + 1 = -z^2

We'll substitute z1 and z2:

z1 + 1 = -z1^2

z2 + 1 = -z2^2

(z1 + 1)^10 + (z2 + 1)^10 = (-z1^2)^10 + (-z2^2)^10

We'll re-write the expression (-z1^2)^10 + (-z2^2)^10:

(-z1^2)^10 + (-z2^2)^10 = (z1^2)^10 + (z2^2)^10

(z1^10)^2 + (z2^10)^2

We'll write z1^10 = z1^(9+1)

z1^(9+1) = z1^9*z1

z1^9*z1 = (z1^3)^2*z1

But z1^3 = 1

z1^10 = 1^2*z1

z1^10 = z1

z2^10 = z2

So, 

(z1^10)^2 + (z2^10)^2 = z1^2 + z2^2

We'll use Viete's relations to express z1 + z2:

z1 + z2 = -b/a

where b and a are the coefficients of the quadratic equation

az^2 + bz + c = 0

In our case, the quadratic is:

z^2 +z +1 = 0

a = 1

b = 1

c = 1

z1 + z2 = -1/1

z1 + z2 = -1

z1*z2 = c/a

z1*z2 = 1

z1^2 + z2^2 = (z1 + z2)^2 - 2z1*z2

(z1 + 1)^10 + (z2 + 1)^10 = (-1)^2 - 2*1

(z1 + 1)^10 + (z2 + 1)^10 = 1 - 2

(z1 + 1)^10 + (z2 + 1)^10 = -1

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