Hi I know is is probably basic but can anyone assist with the following :

Question

Find all real Roots, expressing as whole # and # under radical sign, not as fraction. and factor completely

f(x)= X^3-2x^2-15x+30

I got roots at 2 (x-2) and +/- √5

If someone solves and gets factored as
(x-2) (x^2-15) can you show me line by line how to factor x^2-15?

Answer 1

I noticed that it factors quite easily using grouping

f(x) = x^2(x - 2) - 15(x - 2)
= (x-2)(x^2 - 15)

so if (x-2)(x^2 - 15) = 0
then x-2 = 0 or x^2 - 15 = 0
x = 2

or

x^2 - 15 = 0
x^2 = 15
x = ± √15

x^2 - 15 does not factor using rational numbers
you could look it as a difference of squares, and
(x-√15)(x+√15)
which would produce our other two roots of ± √15

picture of our situation
http://www.wolframalpha.com/input/?i=plot+y+%3D+x%5E3-2x%5E2-15x%2B30

notice it crosses the x - axis at 2 and around -3.9 and + 3.9

√15 = appr 3.9