What are the solutions to the equation y^{16}-y=189 ?

The solutions to the equation y^{16}-y=189 are y\approx-1.38700…,y\approx 1.38828…

Find the zeros of y^{16}-y=189

The zeros of y^{16}-y=189 are y\approx-1.38700…,y\approx 1.38828…

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y^{16} - y = 189

y \approx - 1.38700 \dots, y \approx 1.38828 \dots

Solution steps

y^{16} - y = 189

Move

189

to the left side

y^{16} - y - 189 = 0

Find one solution for

y^{16} - y - 189 = 0

using Newton-Raphson:

y \approx - 1.38700 \dots

Apply long division:

\frac{y ^{16} - y - 189}{y + 1.38700 \dots} = y^{15} - 1.38700 \dots y^{14} + 1.92379 \dots y^{13} - 2.66831 \dots y^{12} + 3.70097 \dots y^{11} - 5.13327 \dots y^{10} + 7.11989 \dots y^{9} - 9.87535 \dots y^{8} + 13.69718 \dots y^{7} - 18.99810 \dots y^{6} + 26.35052 \dots y^{5} - 36.54838 \dots y^{4} + 50.69290 \dots y^{3} - 70.31145 \dots y^{2} + 97.52254 \dots y - 136.26454 \dots

y^{15} - 1.38700 \dots y^{14} + 1.92379 \dots y^{13} - 2.66831 \dots y^{12} + 3.70097 \dots y^{11} - 5.13327 \dots y^{10} + 7.11989 \dots y^{9} - 9.87535 \dots y^{8} + 13.69718 \dots y^{7} - 18.99810 \dots y^{6} + 26.35052 \dots y^{5} - 36.54838 \dots y^{4} + 50.69290 \dots y^{3} - 70.31145 \dots y^{2} + 97.52254 \dots y - 136.26454 \dots \approx 0

Find one solution for

y^{15} - 1.38700 \dots y^{14} + 1.92379 \dots y^{13} - 2.66831 \dots y^{12} + 3.70097 \dots y^{11} - 5.13327 \dots y^{10} + 7.11989 \dots y^{9} - 9.87535 \dots y^{8} + 13.69718 \dots y^{7} - 18.99810 \dots y^{6} + 26.35052 \dots y^{5} - 36.54838 \dots y^{4} + 50.69290 \dots y^{3} - 70.31145 \dots y^{2} + 97.52254 \dots y - 136.26454 \dots = 0

using Newton-Raphson:

y \approx 1.38828 \dots

Apply long division:

\frac{y ^{15} - 1.38700 \dots y ^{14} + 1.92379 \dots y ^{13} - 2.66831 \dots y ^{12} + 3.70097 \dots y ^{11} - 5.13327 \dots y ^{10} + 7.11989 \dots y ^{9} - 9.87535 \dots y ^{8} + 13.69718 \dots y ^{7} - 18.99810 \dots y ^{6} + 26.35052 \dots y ^{5} - 36.54838 \dots y ^{4} + 50.69290 \dots y ^{3} - 70.31145 \dots y ^{2} + 97.52254 \dots y - 136.26454 \dots}{y - 1.38828 \dots} = y^{14} + 0.00127 \dots y^{13} + 1.92555 \dots y^{12} + 0.00490 \dots y^{11} + 3.70778 \dots y^{10} + 0.01416 \dots y^{9} + 7.13956 \dots y^{8} + 0.03636 \dots y^{7} + 13.74768 \dots y^{6} + 0.08754 \dots y^{5} + 26.47205 \dots y^{4} + 0.20227 \dots y^{3} + 50.97372 \dots y^{2} + 0.45441 \dots y + 98.15340 \dots

y^{14} + 0.00127 \dots y^{13} + 1.92555 \dots y^{12} + 0.00490 \dots y^{11} + 3.70778 \dots y^{10} + 0.01416 \dots y^{9} + 7.13956 \dots y^{8} + 0.03636 \dots y^{7} + 13.74768 \dots y^{6} + 0.08754 \dots y^{5} + 26.47205 \dots y^{4} + 0.20227 \dots y^{3} + 50.97372 \dots y^{2} + 0.45441 \dots y + 98.15340 \dots \approx 0

Find one solution for

y^{14} + 0.00127 \dots y^{13} + 1.92555 \dots y^{12} + 0.00490 \dots y^{11} + 3.70778 \dots y^{10} + 0.01416 \dots y^{9} + 7.13956 \dots y^{8} + 0.03636 \dots y^{7} + 13.74768 \dots y^{6} + 0.08754 \dots y^{5} + 26.47205 \dots y^{4} + 0.20227 \dots y^{3} + 50.97372 \dots y^{2} + 0.45441 \dots y + 98.15340 \dots = 0

using Newton-Raphson:

No Solution for

y \in R

The solutions are

y \approx - 1.38700 \dots, y \approx 1.38828 \dots

x^{3} + 2 + x = 0

\frac{2}{3} x^{3} - 18 = 0

x^{3} - 18 x - 8 = 0

e^{0.678} = x^{3}

384 = 3 (r^{8 - 1})

What are the solutions to the equation y^{16}-y=189 ?
The solutions to the equation y^{16}-y=189 are y\approx-1.38700…,y\approx 1.38828…
Find the zeros of y^{16}-y=189
The zeros of y^{16}-y=189 are y\approx-1.38700…,y\approx 1.38828…