What are the solutions to the equation 3x^2-1+x^3-1+x^5=1 ?

The solutions to the equation 3x^2-1+x^3-1+x^5=1 are x\approx 0.82654…

Find the zeros of 3x^2-1+x^3-1+x^5=1

The zeros of 3x^2-1+x^3-1+x^5=1 are x\approx 0.82654…

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3 x^{2} - 1 + x^{3} - 1 + x^{5} = 1

x \approx 0.82654 \dots

Solution steps

3 x^{2} - 1 + x^{3} - 1 + x^{5} = 1

Refine

x^{5} + x^{3} + 3 x^{2} - 2 = 1

Move

1

to the left side

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

Find one solution for

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

using Newton-Raphson:

x \approx 0.82654 \dots

Apply long division:

\frac{x ^{5} + x ^{3} + 3 x ^{2} - 3}{x - 0.82654 \dots} = x^{4} + 0.82654 \dots x^{3} + 1.68318 \dots x^{2} + 4.39122 \dots x + 3.62955 \dots

x^{4} + 0.82654 \dots x^{3} + 1.68318 \dots x^{2} + 4.39122 \dots x + 3.62955 \dots \approx 0

Find one solution for

x^{4} + 0.82654 \dots x^{3} + 1.68318 \dots x^{2} + 4.39122 \dots x + 3.62955 \dots = 0

using Newton-Raphson:

No Solution for

x \in R

The solution is

x \approx 0.82654 \dots

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

10 (4 x^{3} - 15 x^{2} + 16 x - 4) = 0

k^{3} - 3 k^{2} + k + 2 = 0

5 = 3 x^{3} + 4 x^{2} + 6 x + 5

(2 x - 1)^{6} = (2 x - 1)^{8}

What are the solutions to the equation 3x^2-1+x^3-1+x^5=1 ?
The solutions to the equation 3x^2-1+x^3-1+x^5=1 are x\approx 0.82654…
Find the zeros of 3x^2-1+x^3-1+x^5=1
The zeros of 3x^2-1+x^3-1+x^5=1 are x\approx 0.82654…