Factor f (x) = x^4 + x^3 - 8x^2 + 6x + 36 completely. Show your work. Then describe what a graph of this function might look like. Include where the graph would cross the x-axis and the y-axis. How many roots are real numbers and how many roots are imaginary numbers?

Use the rational root theorem to find the possible rational roots. The rational roots theorem says that possible rational roots are + / - factors the constant term (36 here) divided by factors of the leading coefficient (1 here). Possible rational roots are

+ / - 1, 2, 3, 4, 9, 12, 18, 36

Test each zero using the rational root test. To do this, use synthetic division to test the roots. I won't show the work here, but the roots that work are - 2 and - 3. As factors, this is x+2 and x+3.

From the synthetic division, we have x^2-4x+6 left over, which is irreducible.

In factored form:

f (x) = (x+2) (x+3) (x^-4x+6)

You could also use a graphing calculator to find the roots and work backwards to get the factored form too. A TI-89 Titanium would factor the polynomial and give you the above answer.