Question

find all rational roots of f(x).
f(x) = x⁴ - 3x³ - 9x² + 9x + 14
write your answer as a list of simplified values separated by commas, if there is more than one value.

Explanation:

Step1: Apply Rational Root Theorem

Possible rational roots: $\pm1, \pm2, \pm7, \pm14$

Step2: Test $x=1$

$f(1)=1^4 - 3(1)^3 -9(1)^2 +9(1)+14=1-3-9+9+14=12
eq0$

Step3: Test $x=-1$

$f(-1)=(-1)^4 -3(-1)^3 -9(-1)^2 +9(-1)+14=1+3-9-9+14=0$

Step4: Factor out $(x+1)$

Use polynomial division or synthetic division:
$\frac{x^4 -3x^3 -9x^2 +9x+14}{x+1}=x^3-4x^2-5x+14$

Step5: Test $x=2$ on cubic

$f(2)=2^3-4(2)^2-5(2)+14=8-16-10+14=-4
eq0$

Step6: Test $x=-2$ on cubic

$f(-2)=(-2)^3-4(-2)^2-5(-2)+14=-8-16+10+14=0$

Step7: Factor out $(x+2)$ from cubic

$\frac{x^3-4x^2-5x+14}{x+2}=x^2-6x+7$

Step8: Check quadratic for rational roots

Discriminant: $(-6)^2-4(1)(7)=36-28=8$, which is not a perfect square, so no rational roots here.

Answer:

$-1, -2$