Anyone feel like figuring out a math question?

Question

This one is giving me a hard and I'm too tired to really figure it out.

Find all zeros (including complex ones) of this function:

f(x) = 9x^5-94x^3+27x^2+40x-12

Also if you could include the factors too that'd be great. :)

Answer 1

Using the Rational Root Theorem and synthetic division:

-2/3 is one zero. Dividing that one gives you the reduced polynomial
9x^4 - 6x^3 - 90x^2 + 87x - 18

3 is another zero. Dividing the reduced poly. above by (x-3) gives you 9x^3 + 21x^2 - 27x + 6.

2/3 works also. Dividing that into the polynomial reached in the previous step gives you 9x^2 + 27x - 9 = 0

Now solve that one with the quadratic formula.
x = [-27 +/- sqrt(1053)]/18

x = [-27 +/- 9sqrt13]/18
Reduce by 9:

x= (-3 +/- sqrt13)/2. Those are 2 zeros, and the others are +/-(2/3) and 3.