How do you factor f(x)=x312x2+36x32 completely, given that (x-2) is a factor?

1 Answer
Dec 2, 2016

x312x2+36x32=(x2)2(x8)

Explanation:

Knowing that (x2) is a factor you can divide the polynomial:

2.....|.......1.........-12.........+36.........-32
........|...................2..........-20.........+32
........|.......1.........-10..........+16...........0

and find:

x312x2+36x32=(x2)(x210x+16)

We could use the quadratic formula to find the factors of q(x)=(x210x+16)

or go simply by inspection:

1) q(x) has two changes in sign, so it has two positive real roots.
2) The product of the roots is 16
3) the sum of the roots is 10

It's easy to see that x=2 and x=8 are such roots, and thus:

x312x2+36x32=(x2)2(x8)