Consider the function
f(x) = \frac{64 x^3 - 80 x^2+ 24 x}{4 x^3+ 13 x^2 - 32 x+ 15 }

The root(s) of f(x) are:
Enter your answer as a comma-separated list.

The graph of f(x) has hole(s) when x has the following value(s):
Enter your answer as a comma-separated list.

Question

Consider the function
f(x) = \frac{64 x^3 - 80 x^2+ 24 x}{4 x^3+ 13 x^2 - 32 x+ 15 }

The root(s) of f(x) are:
Enter your answer as a comma-separated list.

The graph of f(x) has hole(s) when x has the following value(s):
Enter your answer as a comma-separated list.

ranga · Answer

For the roots, the numerator has to be zero:
64x^3 - 80x^2+ 24x = 0
Solve for x.
You can find the GCF and factor it out of the numerator.
Then you will have a quadratic which you know how to factor or solve.
Being a cubic equation you should get 3 roots.

f(x) is undefined when the denominator is zero.
Set 4 x^3+ 13 x^2 - 32 x+ 15 = 0 and solve for x to find the holes.

anonymous · Answer

I got three roots but they are wrong.

ranga · Answer

64x^3 - 80x^2+ 24x = 0
8x(8x^2 - 10x + 3) = 0
8x = 0 or (8x^2 - 10x + 3) = 0
x = 0 is one root
8x^2 - 10x + 3 = 0
8x^2 - 6x - 4x + 3 = 0
2x(4x - 3) - 1(4x - 3) = 0
(2x - 1)(4x - 3) = 0
x = 1/2 or 3/4
The roots are: x = 0, 1/2, 3/4

anonymous · Answer

That is what I got for the roots but it still says those are wrong :/ Thanks for the help.

ranga · Answer

Oh, I know why it says it is wrong.  The mystery will be solved when you find the holes by setting the denominator to 0.  For those x values f(x) will be undefined and one of the holes happens to be 3/4 which you have to drop it from the roots.
So the roots will be just: 0 and 1/2.

anonymous · Answer

It worked! Thank you!

ranga · Answer

You are welcome.  

Don't forget to do the second part of the problem where you have to find the x values for which the function is not defined (f(x) has "holes").

You need to solve:  4 x^3+ 13 x^2 - 32 x+ 15 = 0