00:02
All right.
00:04
This problem, we're working with a polynomial p of x equals 10x cube minus 7x squared minus 4x plus 1.
00:11
And we want to break it down to its factors.
00:15
We're going to start by talking about descartes's rule of signs, which is named for renรฉ descartes who discovered this pattern.
00:30
And what he stated was that the number of sign changes from term to term in a polynomial will determine the number of possible real roots we will have or less than that by an even integer.
00:44
What we mean by that is this.
00:45
Notice the first term is positive.
00:47
The second term in this polynomial is negative.
00:49
So that's one sign change.
00:51
Negative, negative.
00:52
Then we go from negative back to positive.
00:54
There's a second sign change.
00:57
So there's two changes there, two sign changes.
01:00
That means there's going to be two positive real roots or less than that by an even integer, which 2 minus 2 is going to be 0.
01:06
So there's going to be 2 or 0 positive real roots.
01:14
To determine the number of negative real roots, we have to evaluate p of negative x, which will be 10 times negative x cubed minus 7 times negative x squared, minus 4 times negative x plus 1, which will be negative 10x cubed, minus 7x squared plus 4x plus 1.
01:40
So notice here now we have negative negative change to positive positive.
01:48
So there's one sign change.
01:49
So that means there's going to be one negative real root.
01:59
We can't be less than that by an even integer.
02:01
It wouldn't make sense in this context, so there's going to be one.
02:05
Now to start trying to determine what the roots are, this is where we have to use the rational root theorem.
02:12
Now we're going to talk about possible rational roots.
02:22
To discuss the possible rational roots, it's going to be the factors of the constant term divided by the factors of the lead coefficient.
02:36
So in this case, my numerator, factors of one would be plus or minus one, right? one could either be one times one, or it could be negative one times negative one.
02:45
My denominator will be the factors of 10, so that's going to be plus or minus 1, 2, 5, and 10.
02:53
So i could have one in my numerator and each of these in my denominator.
02:58
So in other words, when i go to list out all my possible rational roots, it could be positive or negative, 1 over 1, 1 over 2, 1 over 5, 1 over 10.
03:10
So notice now i have 1, 2, 3, 4 times 2.
03:13
I have 8 possible rational roots.
03:15
It's a third degree polynomial.
03:16
We can only have 3 roots.
03:18
So these clearly are not all roots.
03:20
These are just possibilities.
03:21
But it gives us a starting point...