Question

List all possible rational zeros for the polynomial below. Find all real zeros of the polynomial below and factor completely. Please show all of your work.
f(x) = 3x^4+17x^3+17x^2-33x-36

Answer 1

I might not answer, but it might help others who want to help you if you showed any work you did so far.

Answer 2

What does the rational root theorem say about the possible real roots of a polynomial?

Answer 3

we need to find all the factors of −36 (coefficient of x0)

and all the factors of 3 (coefficient of x4)

factors of −36:±1,±2,±3,±4,±6,±9,±12,±18,±36

factors of 3:±1,±3

possible rational roots are:
±1/1,±2/1,±3/1,±4/1,±6/1,±9/1,±12/1,±18/1,±36/1

±1/3,±2/3,±3/3,±4/3,±6/3,±9/3,±12/3,±18/3,±36/3

We simplify and remove all the duplicates. The list of all possible rational roots is:
±1/3,±2/3,±1,±4/3,±2,±3,±4,±6,±9,±12,±18,±36

The following are the roots: −3,−1,43

The equation can be written as follows: (x+1)(x+3)(x+3)(3x-4)

Is this correct?????

Answer 4

Maybe show your steps?

Answer 5

that's 4/3 not 43 btw

Answer 6

Step 1: Find a root by trying various x, until f(x) = 0 (they must be factors of -36)
Step 2: Divide the corresponding factor (x - root) into f(x) to get a new polynomial.
Step 3a: If there are still x in your new f(x), go back to step 1.
Step 3b: If there are no x in your new f(x), you have found all the roots (zeroes).

Answer 7

looks good to me, just let me multiply out your equation to be sure...

Answer 8

ok

Answer 9

yup, looks perfect!

Answer 10

Thanks TuringTest :)