What are the solutions to the equation solvefor x,f(x)^4+f(-x)^4=x^{2000} ?

The solutions to the equation solvefor x,f(x)^4+f(-x)^4=x^{2000} are

Find the zeros of solvefor x,f(x)^4+f(-x)^4=x^{2000}

The zeros of solvefor x,f(x)^4+f(-x)^4=x^{2000} are

solve for

x, f (x)^{4} + f (- x)^{4} = x^{2000}

x = 0, x = 4992 499 f, x = - 4992 499 f, x = 4 4992 499 f, x = - 4 4992 499 f

Solution steps

f x^{4} + f (- x)^{4} = x^{2000}

Expand

f x^{4} + f (- x)^{4} : 2 x^{4} f

2 x^{4} f = x^{2000}

Switch sides

x^{2000} = 2 x^{4} f

Move

2 x^{4} f

to the left side

x^{2000} - 2 x^{4} f = 0

Write in the standard form

a_{n} x^{n} + \dots + a_{1} x + f = 0

x^{2000} - 2 f x^{4} = 0

Rewrite the equation with

u = x^{2}, u^{2} = x^{4}

and

u^{1000} = x^{2000}

u^{1000} - 2 f u^{2} = 0

Solve

u^{1000} - 2 f u^{2} = 0 : u = 0, u = 4992 499 f, u = - 4992 499 f

u = 0, u = 4992 499 f, u = - 4992 499 f

Substitute back

u = x^{2},

solve for

x

Solve

x^{2} = 0 : x = 0

Solve

x^{2} = 4992 499 f : x = 4992 499 f, x = - 4992 499 f

Solve

x^{2} = - 4992 499 f : x = 4 4992 499 f, x = - 4 4992 499 f

The solutions are

x = 0, x = 4992 499 f, x = - 4992 499 f, x = 4 4992 499 f, x = - 4 4992 499 f

2 a^{3} - 3 a^{2} + 2 a + 1 = 0

2 x^{5} - 5 x^{4} + 10 x + 3 = 0

1 x^{3} - 7 x + 6 = 0

- 8 x - 12 + x^{3} + x^{2} = 0

0 = x^{3} - x + 1

What are the solutions to the equation solvefor x,f(x)^4+f(-x)^4=x^{2000} ?
The solutions to the equation solvefor x,f(x)^4+f(-x)^4=x^{2000} are
Find the zeros of solvefor x,f(x)^4+f(-x)^4=x^{2000}
The zeros of solvefor x,f(x)^4+f(-x)^4=x^{2000} are