Polynomial inequalities

[The Velociraptors]

I know that the inequality has to be less than or equal to 0 because that’s what the answer said. But I’m having trouble graphing this. Why is three and 1 included if the inequality is less than or equal to 0? They’re both greater than 0. Is it just the set of y values that must be less than 0, or both x and y values less than or equal to 0? If it was only y-values though, then it would be odd how the regions of 3 and 1 aren’t shaded making the domain (-infinity,3). I really don’t understand what is happening, if anyone has any resources or know of any videos that touch on this, that would be extremely helpful.

 

[Steven G]

Very strange problem. Is it complete?!

I will assume that the number line is showing the points where y (NOT x) is less than zero.

Your answer should be in the format of y<=(x+2)^2*(x-1)^2*(x-3) [Not x<=(x+2)^2*(x-1)^2*(x-3)].
The problem is this inequality is <=0 when x<=3. You only want y<=0 when x<=-2.
Try fixing your answer.

 

[The Velociraptors]

Well, it is complete, but has many solutions.
If the y values were less than 0, would it look like this:
But I guess my problem with this is why the interval (-2,3) isn’t shaded. Why is it [1,1] and [3,3]?

 

[lev888]

Well, it is complete, but has many solutions.
If the y values were less than 0, would it look like this:
View attachment 29433But I guess my problem with this is why the interval (-2,3) isn’t shaded. Why is it [1,1] and [3,3]?

You first diagram was perfectly fine, why did you change it? The solution set includes _points_ 1 and 3 - this is where y=0. If you make the graph deep below the x axis to the right of x=-2 you'll include more than 1 and 3.
Can you explain why you decided to square some factors and not the others? How does squaring affect the behavior of the graph?

 

[The Velociraptors]

You first diagram was perfectly fine, why did you change it? The solution set includes _points_ 1 and 3 - this is where y=0. If you make the graph deep below the x axis to the right of x=-2 you'll include more than 1 and 3.
Can you explain why you decided to square some factors and not the others? How does squaring affect the behavior of the graph?

Oh right! I totally forgot that when x=3, y=0, or x=1, y=0; because of the greater than or equal to sign, these coordinates are included. This actually makes a lot of sense, thank you! I decided to square some quantities to try and install a tangent at those points. I was just experimenting with what I could do to satisfy the requirements. It would affect the behavior because it prevents the line from passing through the x-axis.

 

[Steven G]

Well, it is complete, but has many solutions.
If the y values were less than 0, would it look like this:
View attachment 29433But I guess my problem with this is why the interval (-2,3) isn’t shaded. Why is it [1,1] and [3,3]?

You wrote on your own a + sign right above 4 on the number line! You put that because the graph is positive there BUT you want the graph to always be negative or 0.

 

[Steven G]

To OP and helpers, what on earth does the bold on the number line to the left of -2 mean? To me, it would mean y=0, if x<=-2.

 

[Steven G]

The more I think about this problem, the more I think that the polynomial must be piecewise.

 

[The Velociraptors]

To OP and helpers, what on earth does the bold on the number line to the left of -2 mean? To me, it would mean y=0, if x<=-2.

Yes, that is what it means. It also means that the interval is (-infinity,-2], and I’m just supposed to find whether or not that region is positive or negative. There are multiple solutions apparently, I just have to come up with a possible one.

 

[The Velociraptors]

You wrote on your own a + sign right above 4 on the number line! You put that because the graph is positive there BUT you want the graph to always be negative or 0.

Well yeah, I knew that the leading coefficient was positive which is why that specific region would be positive. I just forgot about the part where y=0 when x= 1 and 3 which is what got me confused.

 

[Cubist]

To OP and helpers, what on earth does the bold on the number line to the left of -2 mean? To me, it would mean y=0, if x<=-2.

I think the bold line (and points) on the number line just indicate graphically the domain, all x values, where the following inequality is satisfied

P(x) ≤ 0

There is no y. However, if the graph did show y=P(x) then it would look like OP's dotted line in post #1 (NOT post #3)

 

[Steven G]

So then it would be a piecewise graph (for a different reason then I thought)

 

[Cubist]

OP, your suggested polynomial

$(x+2)^2 (x-1)^2 (x-3) \le 0$

is almost correct. You just need to rethink which factors need to be squared.

 

[The Velociraptors]

OP, your suggested polynomial

$(x+2)^2 (x-1)^2 (x-3) \le 0$

is almost correct. You just need to rethink which factors need to be squared.

I think I realized a trick. Because there are many possible solutions, if I take the shaded region to be below the graph, then the interval [-2,3] would go above the graph. If I take the shaded region to be above the graph, then the opposite would be true for [-2,3]. I’ve found two possible solutions following this:

 

[The Velociraptors]

I think I realized a trick. Because there are many possible solutions, if I take the shaded region to be below the graph, then the interval [-2,3] would go above the graph. If I take the shaded region to be above the graph, then the opposite would be true for [-2,3]. I’ve found two possible solutions following this:

Well actually, the first graph would have a negative sign in front of it because the end behavior is negative. Forgot to add that.