What are the solutions to the equation x^5-2x+1=0 ?

The solutions to the equation x^5-2x+1=0 are x=1,x\approx 0.51879…,x\approx-1.29064…

Find the zeros of x^5-2x+1=0

The zeros of x^5-2x+1=0 are x=1,x\approx 0.51879…,x\approx-1.29064…

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x^{5} - 2 x + 1 = 0

x = 1, x \approx 0.51879 \dots, x \approx - 1.29064 \dots

Solution steps

x^{5} - 2 x + 1 = 0

Factor

x^{5} - 2 x + 1 : (x - 1) (x^{4} + x^{3} + x^{2} + x - 1)

(x - 1) (x^{4} + x^{3} + x^{2} + x - 1) = 0

Using the Zero Factor Principle:

ab = 0

then

a = 0

b = 0

x - 1 = 0 or x^{4} + x^{3} + x^{2} + x - 1 = 0

Solve

x - 1 = 0 : x = 1

Solve

x^{4} + x^{3} + x^{2} + x - 1 = 0 : x \approx 0.51879 \dots, x \approx - 1.29064 \dots

The solutions are

x = 1, x \approx 0.51879 \dots, x \approx - 1.29064 \dots

x^{4} - \frac{1}{8} x = 0

2 x^{3} = 6

x^{3} + 2 x^{2} = 81 x + 162

6 t^{3} - 4 t^{2} = 16 t

3 x^{3} + x^{2} + 12 x + 4 = 0

What are the solutions to the equation x^5-2x+1=0 ?
The solutions to the equation x^5-2x+1=0 are x=1,x\approx 0.51879…,x\approx-1.29064…
Find the zeros of x^5-2x+1=0
The zeros of x^5-2x+1=0 are x=1,x\approx 0.51879…,x\approx-1.29064…