Question

use the rational root theorem to list all possible rational roots of given polynomial.
$p(x)=x^4 + 13x^3 + 55x^2 + 21x - 90$
enter the possible rational roots of $p$ using a comma-separated list.
determine the actual roots of $p$.

Explanation:

Step1: Identify constant & leading coeff

Constant term: $-90$, Leading coefficient: $1$

Step2: List factors of each

Factors of $90$: $\pm1, \pm2, \pm3, \pm5, \pm6, \pm9, \pm10, \pm15, \pm18, \pm30, \pm45, \pm90$
Factors of $1$: $\pm1$

Step3: Apply Rational Root Theorem

Possible roots: $\frac{\text{Factors of } -90}{\text{Factors of } 1} = \pm1, \pm2, \pm3, \pm5, \pm6, \pm9, \pm10, \pm15, \pm18, \pm30, \pm45, \pm90$

Step4: Test possible roots for actual roots

Test $x=1$: $1^4+13(1)^3+55(1)^2+21(1)-90=1+13+55+21-90=0$, so $x=1$ is a root.
Test $x=-2$: $(-2)^4+13(-2)^3+55(-2)^2+21(-2)-90=16-104+220-42-90=0$, so $x=-2$ is a root.
Test $x=-3$: $(-3)^4+13(-3)^3+55(-3)^2+21(-3)-90=81-351+495-63-90=0$, so $x=-3$ is a root.
Test $x=-9$: $(-9)^4+13(-9)^3+55(-9)^2+21(-9)-90=6561-9477+4455-189-90=0$, so $x=-9$ is a root.

Answer:

Possible rational roots: $1, -1, 2, -2, 3, -3, 5, -5, 6, -6, 9, -9, 10, -10, 15, -15, 18, -18, 30, -30, 45, -45, 90, -90$
Actual roots: $1, -2, -3, -9$