What are the solutions to the equation 3x^{36}-140x+137=0 ?

The solutions to the equation 3x^{36}-140x+137=0 are x\approx 1.00000…,x\approx 1.01430…

Find the zeros of 3x^{36}-140x+137=0

The zeros of 3x^{36}-140x+137=0 are x\approx 1.00000…,x\approx 1.01430…

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3 x^{36} - 140 x + 137 = 0

x \approx 1.00000 \dots, x \approx 1.01430 \dots

Solution steps

3 x^{36} - 140 x + 137 = 0

Find one solution for

3 x^{36} - 140 x + 137 = 0

using Newton-Raphson:

x \approx 1.00000 \dots

Apply long division:

\frac{3 x ^{36} - 140 x + 137}{x - 1} = 3 x^{35} + 3 x^{34} + 3 x^{33} + 3 x^{32} + 3 x^{31} + 3 x^{30} + 3 x^{29} + 3 x^{28} + 3 x^{27} + 3 x^{26} + 3 x^{25} + 3 x^{24} + 3 x^{23} + 3 x^{22} + 3 x^{21} + 3 x^{20} + 3 x^{19} + 3 x^{18} + 3 x^{17} + 3 x^{16} + 3 x^{15} + 3 x^{14} + 3 x^{13} + 3 x^{12} + 3 x^{11} + 3 x^{10} + 3 x^{9} + 3 x^{8} + 3 x^{7} + 3 x^{6} + 3 x^{5} + 3 x^{4} + 3 x^{3} + 3 x^{2} + 3 x - 137

3 x^{35} + 3 x^{34} + 3 x^{33} + 3 x^{32} + 3 x^{31} + 3 x^{30} + 3 x^{29} + 3 x^{28} + 3 x^{27} + 3 x^{26} + 3 x^{25} + 3 x^{24} + 3 x^{23} + 3 x^{22} + 3 x^{21} + 3 x^{20} + 3 x^{19} + 3 x^{18} + 3 x^{17} + 3 x^{16} + 3 x^{15} + 3 x^{14} + 3 x^{13} + 3 x^{12} + 3 x^{11} + 3 x^{10} + 3 x^{9} + 3 x^{8} + 3 x^{7} + 3 x^{6} + 3 x^{5} + 3 x^{4} + 3 x^{3} + 3 x^{2} + 3 x - 137 \approx 0

Find one solution for

3 x^{35} + 3 x^{34} + 3 x^{33} + 3 x^{32} + 3 x^{31} + 3 x^{30} + 3 x^{29} + 3 x^{28} + 3 x^{27} + 3 x^{26} + 3 x^{25} + 3 x^{24} + 3 x^{23} + 3 x^{22} + 3 x^{21} + 3 x^{20} + 3 x^{19} + 3 x^{18} + 3 x^{17} + 3 x^{16} + 3 x^{15} + 3 x^{14} + 3 x^{13} + 3 x^{12} + 3 x^{11} + 3 x^{10} + 3 x^{9} + 3 x^{8} + 3 x^{7} + 3 x^{6} + 3 x^{5} + 3 x^{4} + 3 x^{3} + 3 x^{2} + 3 x - 137 = 0

using Newton-Raphson:

x \approx 1.01430 \dots

Apply long division:

\frac{3 x ^{35} + 3 x ^{34} + 3 x ^{33} + 3 x ^{32} + 3 x ^{31} + 3 x ^{30} + 3 x ^{29} + 3 x ^{28} + 3 x ^{27} + 3 x ^{26} + 3 x ^{25} + 3 x ^{24} + 3 x ^{23} + 3 x ^{22} + 3 x ^{21} + 3 x ^{20} + 3 x ^{19} + 3 x ^{18} + 3 x ^{17} + 3 x ^{16} + 3 x ^{15} + 3 x ^{14} + 3 x ^{13} + 3 x ^{12} + 3 x ^{11} + 3 x ^{10} + 3 x ^{9} + 3 x ^{8} + 3 x ^{7} + 3 x ^{6} + 3 x ^{5} + 3 x ^{4} + 3 x ^{3} + 3 x ^{2} + 3 x - 137}{x - 1.01430 \dots} = 3 x^{34} + 6.04292 \dots x^{33} + 9.12940 \dots x^{32} + 12.26003 \dots x^{31} + 15.43547 \dots x^{30} + 18.65634 \dots x^{29} + 21.92331 \dots x^{28} + 25.23702 \dots x^{27} + 28.59815 \dots x^{26} + 32.00738 \dots x^{25} + 35.46539 \dots x^{24} + 38.97288 \dots x^{23} + 42.53056 \dots x^{22} + 46.13916 \dots x^{21} + 49.79939 \dots x^{20} + 53.51199 \dots x^{19} + 57.27773 \dots x^{18} + 61.09735 \dots x^{17} + 64.97162 \dots x^{16} + 68.90133 \dots x^{15} + 72.88728 \dots x^{14} + 76.93027 \dots x^{13} + 81.03110 \dots x^{12} + 85.19062 \dots x^{11} + 89.40966 \dots x^{10} + 93.68907 \dots x^{9} + 98.02972 \dots x^{8} + 102.43248 \dots x^{7} + 106.89824 \dots x^{6} + 111.42791 \dots x^{5} + 116.02239 \dots x^{4} + 120.68262 \dots x^{3} + 125.40953 \dots x^{2} + 130.20408 \dots x + 135.06724 \dots

3 x^{34} + 6.04292 \dots x^{33} + 9.12940 \dots x^{32} + 12.26003 \dots x^{31} + 15.43547 \dots x^{30} + 18.65634 \dots x^{29} + 21.92331 \dots x^{28} + 25.23702 \dots x^{27} + 28.59815 \dots x^{26} + 32.00738 \dots x^{25} + 35.46539 \dots x^{24} + 38.97288 \dots x^{23} + 42.53056 \dots x^{22} + 46.13916 \dots x^{21} + 49.79939 \dots x^{20} + 53.51199 \dots x^{19} + 57.27773 \dots x^{18} + 61.09735 \dots x^{17} + 64.97162 \dots x^{16} + 68.90133 \dots x^{15} + 72.88728 \dots x^{14} + 76.93027 \dots x^{13} + 81.03110 \dots x^{12} + 85.19062 \dots x^{11} + 89.40966 \dots x^{10} + 93.68907 \dots x^{9} + 98.02972 \dots x^{8} + 102.43248 \dots x^{7} + 106.89824 \dots x^{6} + 111.42791 \dots x^{5} + 116.02239 \dots x^{4} + 120.68262 \dots x^{3} + 125.40953 \dots x^{2} + 130.20408 \dots x + 135.06724 \dots \approx 0

Find one solution for

3 x^{34} + 6.04292 \dots x^{33} + 9.12940 \dots x^{32} + 12.26003 \dots x^{31} + 15.43547 \dots x^{30} + 18.65634 \dots x^{29} + 21.92331 \dots x^{28} + 25.23702 \dots x^{27} + 28.59815 \dots x^{26} + 32.00738 \dots x^{25} + 35.46539 \dots x^{24} + 38.97288 \dots x^{23} + 42.53056 \dots x^{22} + 46.13916 \dots x^{21} + 49.79939 \dots x^{20} + 53.51199 \dots x^{19} + 57.27773 \dots x^{18} + 61.09735 \dots x^{17} + 64.97162 \dots x^{16} + 68.90133 \dots x^{15} + 72.88728 \dots x^{14} + 76.93027 \dots x^{13} + 81.03110 \dots x^{12} + 85.19062 \dots x^{11} + 89.40966 \dots x^{10} + 93.68907 \dots x^{9} + 98.02972 \dots x^{8} + 102.43248 \dots x^{7} + 106.89824 \dots x^{6} + 111.42791 \dots x^{5} + 116.02239 \dots x^{4} + 120.68262 \dots x^{3} + 125.40953 \dots x^{2} + 130.20408 \dots x + 135.06724 \dots = 0

using Newton-Raphson:

No Solution for

x \in R

The solutions are

x \approx 1, x \approx 1.01430 \dots

3 x^{3} - 2 x^{2} - 15 x + 10 = 0

x^{7} = 2

x^{7} = 1

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

x^{100} = 100

What are the solutions to the equation 3x^{36}-140x+137=0 ?
The solutions to the equation 3x^{36}-140x+137=0 are x\approx 1.00000…,x\approx 1.01430…
Find the zeros of 3x^{36}-140x+137=0
The zeros of 3x^{36}-140x+137=0 are x\approx 1.00000…,x\approx 1.01430…