What are the solutions to the equation x^4-2x^3-1=0 ?

The solutions to the equation x^4-2x^3-1=0 are x\approx-0.71667…,x\approx 2.10691…

Find the zeros of x^4-2x^3-1=0

The zeros of x^4-2x^3-1=0 are x\approx-0.71667…,x\approx 2.10691…

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

x \approx - 0.71667 \dots, x \approx 2.10691 \dots

Solution steps

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

Find one solution for

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

using Newton-Raphson:

x \approx - 0.71667 \dots

Apply long division:

\frac{x ^{4} - 2 x ^{3} - 1}{x + 0.71667 \dots} = x^{3} - 2.71667 \dots x^{2} + 1.94696 \dots x - 1.39533 \dots

x^{3} - 2.71667 \dots x^{2} + 1.94696 \dots x - 1.39533 \dots \approx 0

Find one solution for

x^{3} - 2.71667 \dots x^{2} + 1.94696 \dots x - 1.39533 \dots = 0

using Newton-Raphson:

x \approx 2.10691 \dots

Apply long division:

\frac{x ^{3} - 2.71667 \dots x ^{2} + 1.94696 \dots x - 1.39533 \dots}{x - 2.10691 \dots} = x^{2} - 0.60975 \dots x + 0.66226 \dots

x^{2} - 0.60975 \dots x + 0.66226 \dots \approx 0

Find one solution for

x^{2} - 0.60975 \dots x + 0.66226 \dots = 0

using Newton-Raphson:

No Solution for

x \in R

The solutions are

x \approx - 0.71667 \dots, x \approx 2.10691 \dots

x^{4} + 7 x^{3} = 0

2 x^{5} - x^{4} - 2 x + 1 = 0

x^{3} - 11 x + 6 = 0

100 (1 + r)^{4} = 84

8 x^{3} + 6 x^{2} - 3 x - 1 = 0

What are the solutions to the equation x^4-2x^3-1=0 ?
The solutions to the equation x^4-2x^3-1=0 are x\approx-0.71667…,x\approx 2.10691…
Find the zeros of x^4-2x^3-1=0
The zeros of x^4-2x^3-1=0 are x\approx-0.71667…,x\approx 2.10691…