Question

select all zeros of the polynomial function.\\(f(x) = x^5 - 20x^3 + 20x^2 - 21x + 20\\)

Explanation:

Step1: Apply Rational Root Theorem

Possible rational roots: $\pm1, \pm2, \pm4, \pm5, \pm10, \pm20$
Test $x=1$: $f(1)=1^5-20(1)^3+20(1)^2-21(1)+20=1-20+20-21+20=0$

Step2: Factor out $(x-1)$

Use polynomial division or synthetic division:
$$\frac{x^5-20x^3+20x^2-21x+20}{x-1}=x^4+x^3-19x^2+x-20$$

Step3: Test roots for quartic

Test $x=4$ on $x^4+x^3-19x^2+x-20$:
$4^4+4^3-19(4)^2+4-20=256+64-304+4-20=0$

Step4: Factor out $(x-4)$

Divide quartic by $(x-4)$:
$$\frac{x^4+x^3-19x^2+x-20}{x-4}=x^3+5x^2+x+5$$

Step5: Factor cubic by grouping

Group terms: $(x^3+5x^2)+(x+5)=x^2(x+5)+1(x+5)=(x^2+1)(x+5)$

Step6: Find all roots

Set each factor to 0:

1. $x-1=0 \implies x=1$
2. $x-4=0 \implies x=4$
3. $x+5=0 \implies x=-5$
4. $x^2+1=0 \implies x=\pm i$

Answer:

$x=-5$, $x=1$, $x=4$, $x=i$, $x=-i$