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

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 2

\[\text{we need to find all the factors of }-36\text{ (coefficient of }x^0) \] \[\text{and all the factors of }3\text{ (coefficient of }x^4)\]
\[\text{factors of }-36 :\pm1, \pm2, \pm 3, \pm4, \pm6, \pm9, \pm 12, \pm18, \pm36\]
\[\text{factors of }3 :\pm1, \pm 3\]
possible rational roots are:
\[\pm\frac{1}{1},\pm\frac{2}{1},\pm\frac{3}{1},\pm\frac{4}{1},\pm\frac{6}{1},\pm\frac{9}{1},\pm\frac{12}{1},\pm\frac{18}{1},\pm\frac{36}{1}\]
\[\pm\frac{1}{3},\pm\frac{2}{3},\pm\frac{3}{3},\pm\frac{4}{3},\pm\frac{6}{3},\pm\frac{9}{3},\pm\frac{12}{3},\pm\frac{18}{3},\pm\frac{36}{3}\]
We simplify and remove all the duplicates. The list of all possible rational roots is:
\[\pm\frac{1}{3},\pm\frac{2}{3},\pm1, \pm\frac{4}{3},\pm2, \pm 3, \pm4, \pm6, \pm9, \pm 12, \pm18, \pm36\]
\[\text{The following are the roots: } -3, -1, \frac{4}{3}\]
\[\text{The equation can be written as follows: (x+1)(x+3)(x+3)(3x-4)}\]

Answer 3

vood- is this the answer or does it need to be simplified further?