The-number-of-common-roots-of-the-equations-x-5-x-3-x-2-1-0-and-x-4-1-0-is-

Question Number 140271 by EnterUsername last updated on 06/May/21

The number of common roots of the equations

x^{5} - x^{3} + x^{2} - 1 = 0 and x^{4} - 1 = 0 is____.

Answered by liberty last updated on 06/May/21

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

\Rightarrow (x + 1) (x - 1) (x + i) (x - i) = 0

\Rightarrow x = \pm 1, x = \pm i

(2) (x - 1) (x^{4} + x^{3} + x + 1) = 0

\Rightarrow (x - 1) (x + 1) (x^{3} + 1) = 0

\Rightarrow (x - 1) (x + 1) (x + 1) (x^{2} - x + 1) = 0

\Rightarrow (x - 1) {(x + 1)}^{2} ({(x - \frac{1}{2})}^{2} + \frac{3}{4}) = 0

\Rightarrow x = \pm 1; x = \pm \frac{\sqrt{3}}{2} i + \frac{1}{2}

the number of common roots is 2

Commented by EnterUsername last updated on 06/May/21

Thank you!