00:01
Okay, so we have an algebra question here.
00:04
We need to find an equation for a polynomial.
00:07
So we're looking for a function.
00:09
And we're going to have a polynomial that has a degree 7.
00:16
And then we want to have the following zeros.
00:22
X equal 10, x equal negative 11, and x equal negative 5.
00:31
We want f of zero to equal negative 5.
00:36
And the only time our function is positive or the only time f of x is greater than zero is on the interval from negative infinity to negative 11.
00:51
So negative infinity to negative 11.
00:54
So if i make kind of a quick sketch, so i know that i want a polynomial degree 7.
01:04
That means that i have an odd order.
01:07
So my end behavior is going to be doing something different.
01:12
As my x values are approaching negative infinity, we can see that our function has to be positive.
01:21
So our x values are going further this way.
01:24
Our function has to be positive, which means we have to start up here somewhere.
01:29
And so my function is going to be going that direction on that left hand side.
01:34
So i know it's going to be going the other direction on the right hand side.
01:38
Now quickly think about a cubic function, just y equals or f of x equal x cubed.
01:49
It looks something like that.
01:51
So what this tells me, because my end behavior here is my x values are approaching negative infinity.
01:59
My function is also approaching negative infinity or it's negative.
02:04
So that tells me that i'm going to have to have a negative coefficient on my leading term because my function over here, my seventh degree function is going up to the positive side.
02:16
Now, i also know that i have zeros at x equal 10, x equal negative 11, and at x equal negative 5.
02:28
So there are my zeros.
02:30
And this tells me this f of zero equal negative 5, that tells me where i cross the y -axis.
02:36
So i cross the y -axis right here at negative 5.
02:41
So i'm going to erase this here to kind of get a quick sketch.
02:47
Okay, the other thing we want to think about, though, is because we are told this is the only interval where our function is positive.
02:56
So all the rest of this function has to be below the x -axis.
03:00
Now, any factor that i have, these give me factors.
03:06
If my multiplicity is odd, i cross the x -axis.
03:11
If my multiplicity is even, i touch the x -axis and go back the other direction.
03:18
So that tells me here on this interval, negative infinity to negative 11, i'm going to cross the x -axis because that's the only place my function is positive.
03:29
So i'm going to come down, i'm going to cross the x -axis there.
03:33
At some point i'm going to turn.
03:35
I'm only going to touch it here because remember i can't go back up to the positive side.
03:40
And then i'm going to cross the y axis here and then i'm going to come here i'm only going to touch it here again because i'm never going to cross over to that positive side again my y values my function values are not going to be positive so one possibility let me make this a little smaller try to get me some room here one possibility and there are more than one but i have this factor x minus 10, and that's the one that gives me this zero.
04:14
Well, i'm going to say, what if this is not a multiplicity of one, but what if it's a multiplicity of three? and that will definitely cross the x -axis because i have an odd exponent here.
04:28
And then i have my other zeros are x plus 11, which gives me this zero.
04:34
That has to have an even order because it has to have.
04:38
Touch the x -axis and go down so that zero has an even multiplicity and then my next zero x plus five that gives me also an even multiplicity so i think i've got my factors mixed up i wrote them in a bad order up here at the top let me just make sure that i write these correctly and go back and explain so my x plus 11 that's the one where i have a cubic function because that's the one where i i cross the x -axis right here.
05:12
My x plus 10 is an even because that one just touches and goes back the other direction...