What are the solutions to the equation r^5+5r^4-1=0 ?

The solutions to the equation r^5+5r^4-1=0 are r\approx 0.64865…,r\approx-0.69420…,r\approx-4.99839…

Find the zeros of r^5+5r^4-1=0

The zeros of r^5+5r^4-1=0 are r\approx 0.64865…,r\approx-0.69420…,r\approx-4.99839…

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r^{5} + 5 r^{4} - 1 = 0

r \approx 0.64865 \dots, r \approx - 0.69420 \dots, r \approx - 4.99839 \dots

Solution steps

r^{5} + 5 r^{4} - 1 = 0

Find one solution for

r^{5} + 5 r^{4} - 1 = 0

using Newton-Raphson:

r \approx 0.64865 \dots

Apply long division:

\frac{r ^{5} + 5 r ^{4} - 1}{r - 0.64865 \dots} = r^{4} + 5.64865 \dots r^{3} + 3.66402 \dots r^{2} + 2.37668 \dots r + 1.54165 \dots

r^{4} + 5.64865 \dots r^{3} + 3.66402 \dots r^{2} + 2.37668 \dots r + 1.54165 \dots \approx 0

Find one solution for

r^{4} + 5.64865 \dots r^{3} + 3.66402 \dots r^{2} + 2.37668 \dots r + 1.54165 \dots = 0

using Newton-Raphson:

r \approx - 0.69420 \dots

Apply long division:

\frac{r ^{4} + 5.64865 \dots r ^{3} + 3.66402 \dots r ^{2} + 2.37668 \dots r + 1.54165 \dots}{r + 0.69420 \dots} = r^{3} + 4.95445 \dots r^{2} + 0.22463 \dots r + 2.22074 \dots

r^{3} + 4.95445 \dots r^{2} + 0.22463 \dots r + 2.22074 \dots \approx 0

Find one solution for

r^{3} + 4.95445 \dots r^{2} + 0.22463 \dots r + 2.22074 \dots = 0

using Newton-Raphson:

r \approx - 4.99839 \dots

Apply long division:

\frac{r ^{3} + 4.95445 \dots r ^{2} + 0.22463 \dots r + 2.22074 \dots}{r + 4.99839 \dots} = r^{2} - 0.04394 \dots r + 0.44429 \dots

r^{2} - 0.04394 \dots r + 0.44429 \dots \approx 0

Find one solution for

r^{2} - 0.04394 \dots r + 0.44429 \dots = 0

using Newton-Raphson:

No Solution for

r \in R

The solutions are

r \approx 0.64865 \dots, r \approx - 0.69420 \dots, r \approx - 4.99839 \dots

(a - 2) (a + 2) - (a - 2)^{2} a = 1

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

r^{3} = \frac{243}{72}

y^{3} = y + 24

d^{4} - 4 d^{3} + 8 d^{2} - 8 d + 4 = 0

What are the solutions to the equation r^5+5r^4-1=0 ?
The solutions to the equation r^5+5r^4-1=0 are r\approx 0.64865…,r\approx-0.69420…,r\approx-4.99839…
Find the zeros of r^5+5r^4-1=0
The zeros of r^5+5r^4-1=0 are r\approx 0.64865…,r\approx-0.69420…,r\approx-4.99839…