We have the function $$x^4 + 4x^3 – 17x^2 -24x + 36 = 0.$$ $x^2 -x – 6$ is a factor of this function. Find all the roots of the polynomial.
So we have $(x-3)(x+2)$, and since it is a quartic we need 2 more solutions. Intuitively I'd say you should divide our quartic by our factors to get another quadratic, but I'm sure there exists a quicker/simpler method. My question is what is the quickest method to do this?
Best Answer
Your intuition is spot on. Polynomial long division is the quickest, at least the standard, route to go here, barring any immediate "sighting" of another root. Divide your quartic by the quadratic $x^2 - x - 6$ to obtain a quadratic quotient, then factor the quotient (or use the quadratic formula) to find any remaining roots, if they exist.
Your quartic divided by the quadratic $x^2 - x - 6$ gives the quadratic quotient $$x^2 + 5x -6 = (x+6)(x - 1)$$
$$x^4 + 4x^3 -17x^2 -24x + 36 = (x-3)(x + 2)(x + 6)(x - 1)$$
So there exist four roots.
You can a find brief tutorial on polynomial long division at the Khan Academy, to better familiarize yourself with it.