00:02
And this problem you're asked to find all the zeros of the polynomial.
00:08
When you have something that's fourth degree, it's going to have four answers.
00:14
It's a pretty tough to factor something in this form.
00:18
But one of the things you can do to help you find the zeros is essentially you do the factors of your constant, divided by your factors of your leading coefficient.
00:29
And so all the factors of 21 would be plus or minus 1.
00:34
Or minus 21, plus or minus 3, and plus and minus 7, and then that's it.
00:44
So there's not that many.
00:46
And then any of those divided by your factor of your leading coefficient would be plus or minus 1.
00:52
So really, the only possible real factors here are going to be the same as the factors of 21.
00:58
But there's quite a few there to check, even still, even though it's fairly simple as far as there's just four, but since there's a plus or minus, that makes it eight, that you would have to check.
01:09
And so one of the things you can do is you can actually toss that into a graphing calculator to give you a little help in deciding where to start.
01:19
And in this case, when i plug this in the graphing calculator, it looks to me like the graph is going to cross the x -axis at negative 3 and it's 7.
01:29
So rather than starting out with 1, plus and minus 1, and then plus and minus 3, since i know one of them is going to be negative 3, i'm going to start there.
01:41
And so we're going to use synthetic division to do this.
01:45
To set up synthetic division, you use all your leading coefficients.
01:49
So i'm going to have a 1, a minus 4, a minus 20, a minus 4, and a minus 21.
02:00
And inside here, we're going to need to put our c.
02:04
And we said one of the zeros is going to be x equals negative 3 based on the graph.
02:10
And since that is a 0, then we'll put in negative 3 here.
02:16
And then we'll just perform synthetic division to see if we get a remainder of 0.
02:21
So bring down your 1, and then negative 3 times 1 is going to be a negative 3.
02:26
Add those, that'll be negative 7.
02:28
Negative 3 times negative 7 would be a positive 21.
02:33
Add those two together, you're going to get 1.
02:35
Negative 3 times 1 would be negative 3.
02:39
Add negative 4, negative 3 to get negative 7.
02:42
Negative 3 times negative 7 would be positive 21, and then you add those two together and get zero for your remainder.
02:51
Okay? and so if you wanted to stop there, i guess you potentially could, but this would be an x to the 3rd minus 7x squared plus x minus 7.
03:01
So we're still in an x to the 3rd.
03:03
So if we go ahead and look at that other hint that we got from our graph, which was another 0 would be 7...