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Okay, here let's talk about the factor theorem and look at a few examples.
A polynomial F (X) has a factor of (X - K) if and only if F (K) = 0.
What this says is that if you have a
factor of (X – K) then if you put K into the function you will get zero. Let's for example use this one. It's an easy one.
If I put the K into this function, I get K - K = 0.
So F (K) = 0. So what it is saying, is that if K when put into the function produces a 0,
then (X - K) is called a factor.
The other way says, if (X - K) is a factor, then K must produce a zero when put into the function.
Let's see this example.
We can factor this function, into two binomials.
These are my factors of the function F.
But what are the zeros? What makes this function equal to zero.
Well, X = 3 and X = -2.
So what we are saying is, when 3 is putting the function I get zero out. Put -2 into the function and we get zero.
So since they are zeros, then we can write the factors. If 3 is a zero then this is a factor and if -2 is a zero then this is a factor.
There are couple more examples.
Factor this into two binomials.
What are the zeros?
-2 and -2. It is a repeated zero.
We considered it to be a single zero. But we have something called multiplicity. It is how many copies of the -2 there are. There are two copies.
So what we say is, that zero is -2 with multiplicity 2.
Next example. It is factored for us.
What is the zero of the first factor?
3.
What is the zero of the second one?
1.
That's what makes both of those factors zero. But the first zero has a multiplicity of 3.
The second one what is the multiplicity? It's whatever power is up here, which is 1.
We don't really need the right multiplicity one but we can if we want.
So that means there are four zeros to this polynomial. It is a fourth degree if you multiply this out.
Over here, what is the zero for this first one here.
If it is just a X outside like that, that means 0 is a zero. Zero times anything will make zero.
The zero for the next factor would be -1.
The zero for the last factor, it's a little tougher to find but set it equal to zero and solve for X.
So we see that X = 1/2, is the other zero.
What are the multiplicities?
Multiplicity of 0, would be 3.
Multiplicity of -1, would be 1.
Multiplicity of 1/2 would be 1.
In our last example, we are going to take this function and write it in factored form.
They tell us the zeros so I'm just going to write them as factors.
So we have (X - 3/2) and (X -2) for the factors.
If I put those numbers in there they create zero.
Now I just have to work with the leading coefficient. When I multiply these two together we have a 13 for the leading coefficient.
Now if I multiply this out I will get the original function back.