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

The solutions to the equation x^5+x^2=1 are x\approx 0.80873…

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

The zeros of x^5+x^2=1 are x\approx 0.80873…

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

x \approx 0.80873 \dots

Solution steps

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

Move

1

to the left side

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

Find one solution for

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

using Newton-Raphson:

x \approx 0.80873 \dots

Apply long division:

\frac{x ^{5} + x ^{2} - 1}{x - 0.80873 \dots} = x^{4} + 0.80873 \dots x^{3} + 0.65404 \dots x^{2} + 1.52894 \dots x + 1.23650 \dots

x^{4} + 0.80873 \dots x^{3} + 0.65404 \dots x^{2} + 1.52894 \dots x + 1.23650 \dots \approx 0

Find one solution for

x^{4} + 0.80873 \dots x^{3} + 0.65404 \dots x^{2} + 1.52894 \dots x + 1.23650 \dots = 0

using Newton-Raphson:

No Solution for

x \in R

The solution is

x \approx 0.80873 \dots

x^{3} = x^{7}

m^{3} = \frac{1}{125}

2 x^{4} + 6 x^{3} - 8 x^{2} - 24 x = 0

2 x (x^{3} - 216) = 0

x^{4} + 2 x^{3} - 14 x^{2} + 2 x + 1 = 0

What are the solutions to the equation x^5+x^2=1 ?
The solutions to the equation x^5+x^2=1 are x\approx 0.80873…
Find the zeros of x^5+x^2=1
The zeros of x^5+x^2=1 are x\approx 0.80873…