00:01
All right, so we have this equation 5 times x squared plus 4 squared times x minus 3 to the third.
00:07
And first we want to find the real zeros in the multiplicity of those zeros.
00:11
So what that means is we're taking the entire equation and setting it equal to 0.
00:16
It's already factored.
00:18
It's written as a set of multiplication problems, which means we just have to check each one of those factors to see when they're equal to 0.
00:26
The 5 is never going to be equal to 0, so we don't have to check that one.
00:29
The next factor we see is x squared plus 4 squared.
00:34
So we're going to set that one equal to 0 and solve it.
00:36
I'm going to go ahead and take the square root of both sides to get rid of that outside squared, and we get x squared plus 4 equals 0.
00:44
Square root of 0 is 0.
00:45
Then i'm going to subtract that 4, and i get x squared equals negative 4.
00:51
To solve for x from here and get rid of that second squared, i would have to take the square root again, and that gives me the square root of a negative number, which is an imaginary number.
01:01
It's not real, and we're only looking for real solutions here, real zeros.
01:05
So that one's not going to give us any real zeros.
01:08
We don't really have to worry about it.
01:10
The next one i have to do is this x squared minus 3 to the third.
01:14
So let's go ahead and do that one over here.
01:15
Just like the other one, i'm going to go ahead and set it equal to 0.
01:19
To get rid of that third, i'm going to take the third root of both sides, and that leaves us with x minus 3 equals 0.
01:26
The third root of 0 is still 0.
01:27
Add 3, and we get x equals 3.
01:31
So that is our real 0.
01:34
Now we just have to say the multiplicity, and that always comes from this exponent, the exponent that's on the factor we're looking at.
01:43
It has a multiplicity of 3.
01:51
So that is your answer to part a.
01:53
It's saying that this 0 kind of repeats itself three times.
01:57
Another way to think about that is x minus 3 to the third is the same thing as x minus 3 times x minus 3 times x minus 3.
02:04
So that factor occurs three times, which means that 0 is going to occur three times as well.
02:10
So your 0 is x equals 3, and it has a multiplicity of 3 because of that third power.
02:16
Now we want to think about our x -intercepts.
02:19
X -intercepts are just another way to say real 0s.
02:22
So the only real 0 that we have is that x equals 3, and we want to know if it's going to cross or just touch it.
02:31
What that means is if it crosses, your polynomial is going to go through the x -axis from the top to the bottom, or from the bottom to the top like those two, and then a touch just kind of bounces.
02:47
It goes down, it touches it, and then goes back up, or vice versa from underneath, up, and then back down.
02:53
But it doesn't actually cross it, it just kind of bounces there.
02:56
You can tell whether it crosses or touches it based on its multiplicity.
03:01
If it is an even multiplicity, it bounces, it just touches.
03:06
If it's an odd multiplicity, it crosses through, it goes down and through.
03:11
So this one at this 0x equals 3 had a multiplicity of 3, which is odd...