So let's see an example.
My example here is: factor this polynomial
over Z mod 7.
Take a minute, pause the video, to try to see if you can factor this.
And now hopefully if you're back, we can give this a shot.
So what should we try to do? Well let's try to find a factor.
We don't have too many techniques at this point,
so let's just guess and check.
Now if I plug in the number - so if I'm guessing and checking,
some simple numbers to plug in should be 0 and plus or minus 1, for sure to start,
and then get a little bit higher, and then hopefully you find one by then.
Here we notice that 1 minus 2 plus 3 minus 4 plus 2 that actually is 0,
and so 1 is a root.
So the sum of the coefficients is 0 if and only if 1 is a root,
it's an interesting little fun exercise to play with yourself, but it's not too hard to prove.
Okay so we have a root, so let's
take that root and let's factor it out.
So we know that x minus 1 is a factor, we can use long division to get that factor out of there,
and so we have f of x is equal to x minus 1 times this polynomial.
So we'll do a long division example later, I don't want to do one now. There's also lots on the
Math 135 Resources Page that you can check out. I think I do at least two there.
So here we have that f of x is equal to this.
Now we look at this new polynomial, and now it's of smaller degree
and we repeat the same process. We say, “Okay well can I factor this polynomial?”
And the answer is yes we can.
The sum of the coefficients is still 0, 1 minus 1 plus 2 minus 2, it's still 0.
So now what we can do is we can either do long division again,
or we can actually group this.
So take the first two terms,
and we have a common factor of x squared which you can pull out,
and we're left with x minus 1. The second two terms, they have a common factor of 2,
which we can pull out and we're left with x minus 1,
and so both these terms have x minus 1, we can factor them out
and we would get this polynomial here.
So we have f of x is equal to x minus 1 squared, times x squared plus 2.
And now we have to look at this polynomial and say, “Well can it be factored?”
Well if it can be factored, again just like what we talked about
in the previous slides, if you can factor a quadratic into irreducible polynomials,
it must be a factor of two - there must be two linear factors
that multiply together to give me x squared plus 2.
But if I have two linear factors and they correspond to roots.
So if this factors over Z mod 7, then there must be roots in Z mod 7.
So just check all the values to see if there are any roots.
There's only 7 values to check, and the operation is done modulo 7, it's pretty quick.
So check 0, 1, 2, 3, 4, 5, 6,
we plug them into x squared plus 2 and we get the values 2, 3, 6, 4, 4, 6, 2.
In particular, none of these are 0.
Actually this last one should be a 3, I apologize,
but yes, nonetheless, none of these are 0
so therefore x squared plus 2 has no root in Z mod 7,
and the above form is completely factorized. And that's it.
So it gives us a nice little example, there's a couple of things to note here though.
So notice that we knew that x equals 1 is a root of this polynomial,
but we didn't know how many times x minus 1 was a factor,
and the way that we did that was we took that root, or we took that factor,
and we factored it out. We got this cubic and then we do this again, okay,
and that gave us an x minus 1 squared. So we saw that 1 was a
root, but we didn't see how many times 1 was a root.
And that leads to the concept of multiplicity.
So we say that this root 1 has multiplicity 2
because x minus 1 squared is a factor of my polynomial,
and that gives us this definition.
So the multiplicity of a root c inside a field F
of a polynomial f of x over the field F
is the largest natural number k such that
x minus c to the power of k is a factor of f of x.
Notice that we can take k to be a natural number because
we already know that c is a root of my polynomial,
and so k must be at least 1.
So for example, the multiplicity of the root 1 in the last example was 2.
Okay, so that's multiplicity of roots.
We've seen an example of how to factor,
but in this example we just basically guessed and hoped that we could find a root.
Now one thing to notice that - I’m going backwards in the slides,
but one thing to note is that in this particular example,
we got lucky that we found a root. It's possible
for a quartic polynomial to factor as the product of two quadratic polynomials,
and in which case how would we do that? And
it turns out this question, we're not going to talk about this much in this class
at least over finite fields, but I mean the question gets a little bit complicated, right?
How do I factor this as the product of two irreducible quadratic polynomials.
It's going to take a little bit of work.
You could try a guess and check method, and that's probably enough for us
but in general, you can sort of see oh this question can be really complicated
Life is pretty good when you have roots but what if this
factored as, let's say, x squared plus 2 squared,
then you might not know what the factors are.
Anyway something to think about.
So what techniques do we have for finding roots? We've already seen an example
and now I'm going to talk about the techniques in general. Well
one thing we can do is we can try to use the Rational Roots Theorem to guess a rational root,
depending on what field were in, right?
So in the previous example, we were in Z mod 7.
Rational Root Theorem doesn't make much sense there, but
we can still try to trial and error and try to guess a root.
That's still does make sense.
We could use the Conjugate Roots Theorem, that we'll see in a little bit.
We've seen an example already, I talked about it I didn't show you, but
we did factoring and grouping. You'll see maybe a couple examples of this later.
We can use long division once we find a root to reduce our problem to something smaller,
and if we get down to a quadratic, life is good.
We know the quadratic formula, and hopefully we can use that to find roots.
So these are the techniques that I would have if I'm trying to factor something.
So keep these in mind when you approach a question
of factoring polynomials.