Question

use the rational root theorem to list all possible rational roots of the given polynomial.
$p(x)=x^4 - 9x^3 + 39x^2 - 89x + 78$
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: $78$, Leading coefficient: $1$

Step2: List factors of each

Factors of $78$: $\pm1, \pm2, \pm3, \pm6, \pm13, \pm26, \pm39, \pm78$
Factors of $1$: $\pm1$

Step3: Apply Rational Root Theorem

Possible roots: $\pm\frac{1}{1}, \pm\frac{2}{1}, \pm\frac{3}{1}, \pm\frac{6}{1}, \pm\frac{13}{1}, \pm\frac{26}{1}, \pm\frac{39}{1}, \pm\frac{78}{1}$

Step4: Test possible roots for actual roots

Test $x=2$: $2^4 -9(2)^3 +39(2)^2 -89(2)+78=16-72+156-178+78=0$
Test $x=3$: $3^4 -9(3)^3 +39(3)^2 -89(3)+78=81-243+351-267+78=0$
Factor polynomial: $(x-2)(x-3)(x^2-4x+13)=0$
Solve quadratic: $x=\frac{4\pm\sqrt{16-52}}{2}=2\pm3i$ (not rational)

Answer:

Possible rational roots: $-78, -39, -26, -13, -6, -3, -2, -1, 1, 2, 3, 6, 13, 26, 39, 78$
Actual roots: $2, 3, 2+3i, 2-3i$