What are the solutions to the equation (x^3+1)^3-16x+8=0 ?

The solutions to the equation (x^3+1)^3-16x+8=0 are x=1,x\approx 0.61803…,x\approx-1.61803…

Find the zeros of (x^3+1)^3-16x+8=0

The zeros of (x^3+1)^3-16x+8=0 are x=1,x\approx 0.61803…,x\approx-1.61803…

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(x^{3} + 1)^{3} - 16 x + 8 = 0

x = 1, x \approx 0.61803 \dots, x \approx - 1.61803 \dots

Solution steps

(x^{3} + 1)^{3} - 16 x + 8 = 0

Expand

(x^{3} + 1)^{3} - 16 x + 8 : x^{9} + 3 x^{6} + 3 x^{3} - 16 x + 9

x^{9} + 3 x^{6} + 3 x^{3} - 16 x + 9 = 0

Factor

x^{9} + 3 x^{6} + 3 x^{3} - 16 x + 9 : (x - 1) (x^{8} + x^{7} + x^{6} + 4 x^{5} + 4 x^{4} + 4 x^{3} + 7 x^{2} + 7 x - 9)

(x - 1) (x^{8} + x^{7} + x^{6} + 4 x^{5} + 4 x^{4} + 4 x^{3} + 7 x^{2} + 7 x - 9) = 0

Using the Zero Factor Principle:

ab = 0

then

a = 0

b = 0

x - 1 = 0 or x^{8} + x^{7} + x^{6} + 4 x^{5} + 4 x^{4} + 4 x^{3} + 7 x^{2} + 7 x - 9 = 0

Solve

x - 1 = 0 : x = 1

Solve

x^{8} + x^{7} + x^{6} + 4 x^{5} + 4 x^{4} + 4 x^{3} + 7 x^{2} + 7 x - 9 = 0 : x \approx 0.61803 \dots, x \approx - 1.61803 \dots

The solutions are

x = 1, x \approx 0.61803 \dots, x \approx - 1.61803 \dots

5 x^{3} + 7 x^{2} - 11 x + 3 = 0

z^{4} = - 2 + 23 i

x (x + 7) (x^{2} + 49) = 0

x, x^{3} + y^{3} = - 9

x^{3} + x^{2} + x + 3 = 0

What are the solutions to the equation (x^3+1)^3-16x+8=0 ?
The solutions to the equation (x^3+1)^3-16x+8=0 are x=1,x\approx 0.61803…,x\approx-1.61803…
Find the zeros of (x^3+1)^3-16x+8=0
The zeros of (x^3+1)^3-16x+8=0 are x=1,x\approx 0.61803…,x\approx-1.61803…