00:01
In this problem, our job is to determine the complete factorization of the polynomial x cubed plus 2x squared minus 23x minus 60.
00:10
We're given four choices.
00:12
They all involve the factors x plus 3 or x minus 3, x plus 4 or x minus 4, x plus 5 or x minus 5.
00:20
What we want to do is use a theorem that says that the linear polynomial x minus a divides f of x precisely when f of a is 0.
00:38
This is a very nice shortcut to see when a linear factor divides a polynomial.
00:46
So to find out whether or not x minus 3 divides x, we need to substitute 3.
00:55
So first of all, let's let this polynomial, let me come back up here and indicate this, we're going to let this polynomial be f.
01:04
So f will be the original x cubed plus 2x squared minus 23x minus 60.
01:17
So to answer our first question, is x minus 3 a factor, we need to calculate f.
01:24
So to calculate f of 3 we'll need to cube 3 plus 2 times 3 squared minus 23 times 3 minus 60.
01:36
And we can see that that's going to be negative so that's not going to be 0.
01:41
So we can rule out x minus 3 as a factor.
01:47
Well let's continue that process.
01:49
If x minus 3 is not a factor then maybe x plus 3 is a factor.
01:53
So let's now try x plus 3, see if that's going to work.
02:00
We will substitute f of, we will substitute negative 3 into f.
02:04
So f of negative 3, we need to cube negative 3 and so on.
02:12
And then when we simplify this expression, we see that we get 0.
02:19
So that's good.
02:21
So since we got 0, that means that x x plus 3 is a factor of f of x, so we have one of the factors already.
02:30
Let's do the same thing where we're going to try x minus 4x plus 4, x minus 5x plus 5.
02:39
So will x minus 4 be a factor of f of x? well, it turns out that f of 4 is not 0, it's negative 56, so this does not work...