Question

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?

Answer 1

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.