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

The solutions to the equation (x-1)^5=x^2(x-3)+x are x\approx 0.43965…

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

The zeros of (x-1)^5=x^2(x-3)+x are x\approx 0.43965…

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

x \approx 0.43965 \dots

Solution steps

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

Expand

(x - 1)^{5} : x^{5} - 5 x^{4} + 10 x^{3} - 10 x^{2} + 5 x - 1

Expand

x^{2} (x - 3) + x : x^{3} - 3 x^{2} + x

x^{5} - 5 x^{4} + 10 x^{3} - 10 x^{2} + 5 x - 1 = x^{3} - 3 x^{2} + x

Move

x

to the left side

x^{5} - 5 x^{4} + 10 x^{3} - 10 x^{2} + 4 x - 1 = x^{3} - 3 x^{2}

Move

3 x^{2}

to the left side

x^{5} - 5 x^{4} + 10 x^{3} - 7 x^{2} + 4 x - 1 = x^{3}

Move

x^{3}

to the left side

x^{5} - 5 x^{4} + 9 x^{3} - 7 x^{2} + 4 x - 1 = 0

Find one solution for

x^{5} - 5 x^{4} + 9 x^{3} - 7 x^{2} + 4 x - 1 = 0

using Newton-Raphson:

x \approx 0.43965 \dots

Apply long division:

\frac{x ^{5} - 5 x ^{4} + 9 x ^{3} - 7 x ^{2} + 4 x - 1}{x - 0.43965 \dots} = x^{4} - 4.56034 \dots x^{3} + 6.99504 \dots x^{2} - 3.92462 \dots x + 2.27453 \dots

x^{4} - 4.56034 \dots x^{3} + 6.99504 \dots x^{2} - 3.92462 \dots x + 2.27453 \dots \approx 0

Find one solution for

x^{4} - 4.56034 \dots x^{3} + 6.99504 \dots x^{2} - 3.92462 \dots x + 2.27453 \dots = 0

using Newton-Raphson:

No Solution for

x \in R

The solution is

x \approx 0.43965 \dots

x^{5} - x - 2 = 0

x^{6} - 26 x^{3} = 27

2 x^{4} + 3 x^{3} + 5 x^{2} + 2 x + 6 = 0

0 = y^{3} + 24 y^{2} - 180 y + 432

35 - x^{3} = 5000 - 1000 x

What are the solutions to the equation (x-1)^5=x^2(x-3)+x ?
The solutions to the equation (x-1)^5=x^2(x-3)+x are x\approx 0.43965…
Find the zeros of (x-1)^5=x^2(x-3)+x
The zeros of (x-1)^5=x^2(x-3)+x are x\approx 0.43965…