What are the solutions to the equation s^5-s^3-s^2+1=0 ?

The solutions to the equation s^5-s^3-s^2+1=0 are s=-1,s=1,s=-1/2+i(sqrt(3))/2 ,s=-1/2-i(sqrt(3))/2

Find the zeros of s^5-s^3-s^2+1=0

The zeros of s^5-s^3-s^2+1=0 are s=-1,s=1,s=-1/2+i(sqrt(3))/2 ,s=-1/2-i(sqrt(3))/2

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s^{5} - s^{3} - s^{2} + 1 = 0

s = - 1, s = 1, s = - \frac{1}{2} + i \frac{3}{2}, s = - \frac{1}{2} - i \frac{3}{2}

Solution steps

s^{5} - s^{3} - s^{2} + 1 = 0

Factor

s^{5} - s^{3} - s^{2} + 1 : (s + 1) (s - 1)^{2} (s^{2} + s + 1)

(s + 1) (s - 1)^{2} (s^{2} + s + 1) = 0

Using the Zero Factor Principle:

ab = 0

then

a = 0

b = 0

s + 1 = 0 or s - 1 = 0 or s^{2} + s + 1 = 0

Solve

s + 1 = 0 : s = - 1

Solve

s - 1 = 0 : s = 1

Solve

s^{2} + s + 1 = 0 : s = - \frac{1}{2} + i \frac{3}{2}, s = - \frac{1}{2} - i \frac{3}{2}

The solutions are

s = - 1, s = 1, s = - \frac{1}{2} + i \frac{3}{2}, s = - \frac{1}{2} - i \frac{3}{2}

a^{5} = 7

x^{4} + x^{3} + x + 1 = 4 x^{2}

x^{4} - 18 x^{2} + 32 = 0

x^{3} - 5 x^{2} + 6 x - 1 = 0

x^{3} + x^{2} - 21 x - 45 = 0

What are the solutions to the equation s^5-s^3-s^2+1=0 ?
The solutions to the equation s^5-s^3-s^2+1=0 are s=-1,s=1,s=-1/2+i(sqrt(3))/2 ,s=-1/2-i(sqrt(3))/2
Find the zeros of s^5-s^3-s^2+1=0
The zeros of s^5-s^3-s^2+1=0 are s=-1,s=1,s=-1/2+i(sqrt(3))/2 ,s=-1/2-i(sqrt(3))/2