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

The solutions to the equation x+x^2+x^3+x^4=1 are x\approx 0.51879…,x\approx-1.29064…

Find the zeros of x+x^2+x^3+x^4=1

The zeros of x+x^2+x^3+x^4=1 are x\approx 0.51879…,x\approx-1.29064…

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x + x^{2} + x^{3} + x^{4} = 1

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

Solution steps

x + x^{2} + x^{3} + x^{4} = 1

Move

1

to the left side

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

Write in the standard form

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

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

Find one solution for

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

using Newton-Raphson:

x \approx 0.51879 \dots

Apply long division:

\frac{x ^{4} + x ^{3} + x ^{2} + x - 1}{x - 0.51879 \dots} = x^{3} + 1.51879 \dots x^{2} + 1.78793 \dots x + 1.92756 \dots

x^{3} + 1.51879 \dots x^{2} + 1.78793 \dots x + 1.92756 \dots \approx 0

Find one solution for

x^{3} + 1.51879 \dots x^{2} + 1.78793 \dots x + 1.92756 \dots = 0

using Newton-Raphson:

x \approx - 1.29064 \dots

Apply long division:

\frac{x ^{3} + 1.51879 \dots x ^{2} + 1.78793 \dots x + 1.92756 \dots}{x + 1.29064 \dots} = x^{2} + 0.22814 \dots x + 1.49348 \dots

x^{2} + 0.22814 \dots x + 1.49348 \dots \approx 0

Find one solution for

x^{2} + 0.22814 \dots x + 1.49348 \dots = 0

using Newton-Raphson:

No Solution for

x \in R

The solutions are

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

x, x^{3} - 7 x^{2} = 7 - x

x^{4} - 4 x^{3} + 4 x^{2} - 36 x - 45 = 0

12 x^{3} - 6 x^{2} = 0

x^{3} - 9 x^{2} + 31 x - 39

4 x^{3} - 18 x^{2} + 24 x = 0

What are the solutions to the equation x+x^2+x^3+x^4=1 ?
The solutions to the equation x+x^2+x^3+x^4=1 are x\approx 0.51879…,x\approx-1.29064…
Find the zeros of x+x^2+x^3+x^4=1
The zeros of x+x^2+x^3+x^4=1 are x\approx 0.51879…,x\approx-1.29064…