Question

1. find the roots of $p(x)=x^{4}+x^{3}-x^{2}+5x - 30$

Explanation:

Step1: Use Rational Root Theorem

Possible rational roots: $\pm1, \pm2, \pm3, \pm5, \pm6, \pm10, \pm15, \pm30$
Test $x=2$: $2^4+2^3-2^2+5(2)-30=16+8-4+10-30=0$
Test $x=-3$: $(-3)^4+(-3)^3-(-3)^2+5(-3)-30=81-27-9-15-30=0$

Step2: Factor polynomial

$P(x)=(x-2)(x+3)(x^2+0x+5)=(x-2)(x+3)(x^2+5)$

Step3: Solve quadratic factor

Set $x^2+5=0$: $x^2=-5$ so $x=\pm\sqrt{5}i$

Answer:

The roots are $2$, $-3$, $\sqrt{5}i$, and $-\sqrt{5}i$