00:01
Okay, let's suppose we're given a polynomial inequality.
00:04
How are we going to solve this polynomial inequality? well, what a lot of people would like to do is i see that each term has x squared, so i'm just going to divide by x squared each term.
00:13
Technically, i could do that, but i want to be careful whenever i move x squared terms.
00:19
Why is that? i could just do that in an equation and it's okay.
00:23
Well, the differences, when i have an inequality, note that i, when i multiply or divide by a negative number, it can flip the inequality.
00:32
And sometimes when i'm moving around x terms, i'm not sure if they are completely positive or negative all the time.
00:39
So first thing i'm going to do with a polynomial inequality is just get all the terms to one side.
00:44
So i'm going to subtract 12x squared from both sides.
00:48
Technically set it equal to 0.
00:50
So i have 2x the 4th plus 2x the 3rd minus 12x squared.
00:56
And then what am i going to do? well, now i'm just going to, since i have all my terms on one side, i'm going to deal with this like i am solving.
01:06
I'm setting the polynomial equal to zero and just solving for x.
01:10
So first thing i'm going to do is just solve for x and find some critical points here.
01:15
Notice that i could pull out a greatest common factor.
01:17
They're all divisible by two, and they all have at least two x's.
01:21
And if i divide by that to each term, i'm going to get so.
01:25
I can continue to factor a quadratic equation rate.
01:29
1 multiplies to negative 6, but adds to positive 1.
01:34
Well, 3 and 2, and then to positive 1 plus 3 minus 2.
01:39
And now, 0 product property said each of these equal to 0.
01:43
If i divide by 2 and take the square root, i'll just get x equals 0.
01:48
X plus 3 equals 0, subtract 3.
01:51
My polynomial is 0 at x equals negative 3, and then my polynomial also zero at x equals two.
01:59
Okay, these are my critical points.
02:01
What do i do with my critical points? well, now i can just create intervals from them.
02:05
So i could start at negative infinity to negative three.
02:09
And from these intervals, what i could do is i could figure out where my polynomial is changing.
02:16
So across, it becomes zero, and then either goes from positive to negative, negative to positive, or it stays the same and just kind of touch to the x -axis.
02:25
So i'm gonna plug these all in, either to my original function when i got it equal to zero, or i could plug it into my factored function.
02:32
I'm going to do my factored function, just to make things a little simpler.
02:36
So i could pick a test point at each interval.
02:39
It does not matter what the test point is.
02:42
So i'm going to pick something that's easy as possible...