What are the solutions to the equation x^{23}-2x^{13}-15=0 ?

The solutions to the equation x^{23}-2x^{13}-15=0 are x\approx 1.15626…

Find the zeros of x^{23}-2x^{13}-15=0

The zeros of x^{23}-2x^{13}-15=0 are x\approx 1.15626…

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x^{23} - 2 x^{13} - 15 = 0

x \approx 1.15626 \dots

Solution steps

x^{23} - 2 x^{13} - 15 = 0

Find one solution for

x^{23} - 2 x^{13} - 15 = 0

using Newton-Raphson:

x \approx 1.15626 \dots

Apply long division:

\frac{x ^{23} - 2 x ^{13} - 15}{x - 1.15626 \dots} = x^{22} + 1.15626 \dots x^{21} + 1.33695 \dots x^{20} + 1.54588 \dots x^{19} + 1.78745 \dots x^{18} + 2.06677 \dots x^{17} + 2.38975 \dots x^{16} + 2.76319 \dots x^{15} + 3.19499 \dots x^{14} + 3.69427 \dots x^{13} + 2.27157 \dots x^{12} + 2.62655 \dots x^{11} + 3.03699 \dots x^{10} + 3.51158 \dots x^{9} + 4.06033 \dots x^{8} + 4.69484 \dots x^{7} + 5.42849 \dots x^{6} + 6.27680 \dots x^{5} + 7.25767 \dots x^{4} + 8.39181 \dots x^{3} + 9.70319 \dots x^{2} + 11.21950 \dots x + 12.97276 \dots

x^{22} + 1.15626 \dots x^{21} + 1.33695 \dots x^{20} + 1.54588 \dots x^{19} + 1.78745 \dots x^{18} + 2.06677 \dots x^{17} + 2.38975 \dots x^{16} + 2.76319 \dots x^{15} + 3.19499 \dots x^{14} + 3.69427 \dots x^{13} + 2.27157 \dots x^{12} + 2.62655 \dots x^{11} + 3.03699 \dots x^{10} + 3.51158 \dots x^{9} + 4.06033 \dots x^{8} + 4.69484 \dots x^{7} + 5.42849 \dots x^{6} + 6.27680 \dots x^{5} + 7.25767 \dots x^{4} + 8.39181 \dots x^{3} + 9.70319 \dots x^{2} + 11.21950 \dots x + 12.97276 \dots \approx 0

Find one solution for

x^{22} + 1.15626 \dots x^{21} + 1.33695 \dots x^{20} + 1.54588 \dots x^{19} + 1.78745 \dots x^{18} + 2.06677 \dots x^{17} + 2.38975 \dots x^{16} + 2.76319 \dots x^{15} + 3.19499 \dots x^{14} + 3.69427 \dots x^{13} + 2.27157 \dots x^{12} + 2.62655 \dots x^{11} + 3.03699 \dots x^{10} + 3.51158 \dots x^{9} + 4.06033 \dots x^{8} + 4.69484 \dots x^{7} + 5.42849 \dots x^{6} + 6.27680 \dots x^{5} + 7.25767 \dots x^{4} + 8.39181 \dots x^{3} + 9.70319 \dots x^{2} + 11.21950 \dots x + 12.97276 \dots = 0

using Newton-Raphson:

No Solution for

x \in R

The solution is

x \approx 1.15626 \dots

x^{3} - 16 x^{2} + 16 x + 6 = 0

(x^{2} - 1) (x - 2) = 2

8 x^{3} - 12 x = 0

x^{12} - 8 x^{2} = 9

- \frac{40}{3} t^{3} + 8000 t = 5440

What are the solutions to the equation x^{23}-2x^{13}-15=0 ?
The solutions to the equation x^{23}-2x^{13}-15=0 are x\approx 1.15626…
Find the zeros of x^{23}-2x^{13}-15=0
The zeros of x^{23}-2x^{13}-15=0 are x\approx 1.15626…