What are the solutions to the equation FUNCTION_MANY#domain of f(f)=f^8 ?

The solutions to the equation FUNCTION_MANY#domain of f(f)=f^8 are f=0,f=1

Find the zeros of FUNCTION_MANY#domain of f(f)=f^8

The zeros of FUNCTION_MANY#domain of f(f)=f^8 are f=0,f=1

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f = f^{8}

f = 0, f = 1

Solution steps

f = f^{8}

Switch sides

f^{8} = f

Move

f

to the left side

f^{8} - f = 0

Factor

f^{8} - f : f (f - 1) (f^{6} + f^{5} + f^{4} + f^{3} + f^{2} + f + 1)

f (f - 1) (f^{6} + f^{5} + f^{4} + f^{3} + f^{2} + f + 1) = 0

Using the Zero Factor Principle:

ab = 0

then

a = 0

b = 0

f = 0 or f - 1 = 0 or f^{6} + f^{5} + f^{4} + f^{3} + f^{2} + f + 1 = 0

Solve

f - 1 = 0 : f = 1

Solve

f^{6} + f^{5} + f^{4} + f^{3} + f^{2} + f + 1 = 0 :

No Solution for

f \in R

The solutions are

f = 0, f = 1

What are the solutions to the equation FUNCTION_MANY#domain of f(f)=f^8 ?
The solutions to the equation FUNCTION_MANY#domain of f(f)=f^8 are f=0,f=1
Find the zeros of FUNCTION_MANY#domain of f(f)=f^8
The zeros of FUNCTION_MANY#domain of f(f)=f^8 are f=0,f=1