Question

how many zeros does the following function have? f(x)=x⁵ - 5x³ + x

Explanation:

Step1: Factor out common term

$f(x) = x(x^4 - 3x^2 + 1)$

Step2: Set function to 0

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

Step3: Solve for first root

$x = 0$

Step4: Solve quartic as quadratic

Let $u = x^2$, so $u^2 - 3u + 1 = 0$

Step5: Apply quadratic formula

$u = \frac{3 \pm \sqrt{9 - 4}}{2} = \frac{3 \pm \sqrt{5}}{2}$

Step6: Find x values from u

$x = \pm\sqrt{\frac{3 + \sqrt{5}}{2}}, \pm\sqrt{\frac{3 - \sqrt{5}}{2}}$

Step7: Count distinct roots

Roots: $0, \sqrt{\frac{3 + \sqrt{5}}{2}}, -\sqrt{\frac{3 + \sqrt{5}}{2}}, \sqrt{\frac{3 - \sqrt{5}}{2}}, -\sqrt{\frac{3 - \sqrt{5}}{2}}$ (total 5)

Answer:

5