Finding the roots of the first derivative?

Question

I got to this point
4x^3 - 16x^2 + 12x
------------------------
(x-1)

put not to sure where to go from there to find the two roots

Sorry its 18x^2 not 16

yes we are allowed to factor the numerator

Answer 1

Can you factor the numerator?

I guess not, but UMR math major ignores your denominator in the original problem.

Answer 2

You can take a 2x out to get

2x[2x^2 - 9x + 6]

then use quadratic formula

-b (+/-) sqrt(b^2-4ac) all over 2a

9 +/- sqrt(81-(4)(2)(6) all over 4

9 (+/-) sqrt(33) all over 4.

Your roots are 9/4 plus or minus root(33)/4 (and zero, dont forget that 2x we took out)

EDIT: Right, I ignore the denominator because as long as the numerator comes up zero, the entire term will be zero. Also, I noted to myself that x=1 cannot possibly be one since it makes the denominator zero.
(Mathematica confirms that my above answers are correct.)

Hope that helped.

UMRmathmajor