Question

How to list all factors and their multiplicity of a polynomial?

Answer 1

\[x^{6}-9x^{5}+12x^{4}+51x^{3}-39x^{2}-60x-100\]

Answer 2

Whoa. That's quite a polynomial.

Answer 3

I know... PreCalc is killing me :(

Answer 4

So one way to do it is to factor the polynomial completely.
That way you can read out the factors, and the multiplicities easily.

Answer 5

Yea. But this one looks complicated to factor.

Answer 6

okay I've only ever done that while having something to divivde it by, like "x-2"

Answer 7

Let's try to factor by grouping and see if it works.

Answer 8

Hmmm..I don't think that will work.

Answer 9

So there's something called the Descartes' Rule of Signs that you can use to determine the number of real zeros.

Answer 10

something about P/Q maybe?

Answer 11

Perhaps we can use that to narrow down the possibilities.

Answer 12

Hmmm...i'm not familiar with the P and Q. Can you describe it and it may be familiar.

Answer 13

you take the first coefficient, P, and the last number and the last, Q, and it gives you all the numbers that can be used to get the factors. I probably butchered that really badly...

Answer 14

Ohh, Ok - That's called the rational roots test. That will give you the number of rational solutions.

Answer 15

Or, I should say, possible roots. But you would have to test them all out.

Answer 16

We can try doing that though.

Answer 17

So p = 1
ANd q = -100

Answer 18

So q = +-1, +-2, +-5, +-10, +-20, +-50, +-100
So those are all the possible q's. Which are the factors of the last term.

Answer 19

okay I'm with you. When I did that I got a bunch of +-numbers

Answer 20

+-1, +-2, +-4, +-5, +-10, +-20, +-25, +-50, +-100

Answer 21

Oh, actually... You are right!
I forgot the +-4, and +-25

Answer 22

mathway told me the only ones that work are 2,-2, and 5 haha

Answer 23

but I have no idea how it got that

Answer 24

LOL. =]

So the POSSIBLE rational roots to this are q/p, right?
Since p is just 1, then q is all the possible rational roots.

So the only way I can see you factoring this is to test each root out.
And see if it divides the polynomial.

For example, for the root -5, you would try to divide the polynomial by x+5, and see if it divides.

Answer 25

That means that those are roots.
Using this test, the only way is by trail and error.
If you were to divide the polynomial by (x-2), (x+2), or (x-5), it would divide evenly.
For any other factor, you have a remainder.

Answer 26

But, yeah, there's no quick way to just determine that, as far as I know.

Answer 27

okay so those would be the factors right? x-2, x+2, and x+5

Answer 28

If the root is x=5, it would be (x-5), not (x+5)
Because (x-5) = 0 gives you x = 5