What factors determine the exponent of a quartic polynomial function?

[Nelo]

Homework Statement

(There is a comparison question I have, so i'll post my question with these)

8) Each of the following polonomials has x-intercepts of -6, 5 , and 0. Determien the appropriate equation for each. Then, sketch a graph of the function.

b) A quartic function that extends from q3 to q4.


So, I understand the graph , and wrote it to be ... -x(x+6) (x-5)

Since its a quartic function it has a exponent of 4, which is positive so a<0 means it goes q3 to q4

However, in the answer booklet it says -x^2 (x+6) (x-5) . How do i know when its going to be x^2? Is it because its an even function and it is symmetrical at the orgin with a y int of 0 , that it will be x^2? Or something else?

Tl:dr , why is that -x^2 instead of -x , what do i look for inorder to determine the diference?

Let me provide an example from the text.

http://i56.tinypic.com/wml21g.jpg

So number 7.b would be the graphical representation, i assume its x² because its on the vertex, but how is this determined? from the text.. and from the graph?

Homework Equations

The Attempt at a Solution

My attempt at the solution was.. -x (x+6) (x-5)

The answer is -x^2 (x+6) (x-5)

Thank you for your help!

 

[Nelo]

any1?

 

[gb7nash]

extends from q3 to q4.

I'm not familiar with this notation. What is q3? q4?

edit: I read the picture. I still don't know what it means to extend from quadrant 3 to 4. Does this mean that the function is nonpositive?

 

[Nelo]

It means that the endpoints extend from quadrant 3 to quadrant 4 . They give that information for you to determine if (a) should be < 0 or > 0 , in correlation with even/odd exponent. Does that help? It means endpoints , ie the arrows at where the graph extends out to

 

[Nelo]

Anyone have a answer to my question though... "So number 7.b would be the graphical representation, i assume its x2 because its on the vertex, but how is this determined? from the text.. and from the graph?"

 

[vela]

The Attempt at a Solution

My attempt at the solution was.. -x (x+6) (x-5)

The answer is -x^2 (x+6) (x-5)

Thank you for your help!

Your solution clearly isn't correct because it's a cubic, not a quartic.

Your book seems a little sloppy. For example, the problem statement merely says there are zeros at x=-6, 0, and 5, but it doesn't say those are the only zeros. There is an infinite number of quartics that have those zeros and extend from quadrant III to IV. If those are the only three roots, there are still 3 possible quartics that satisfy the conditions.

 

[vela]

Anyone have a answer to my question though... "So number 7.b would be the graphical representation, i assume its x2 because its on the vertex, but how is this determined? from the text.. and from the graph?"

Problem 8b you mean? The graph from 7b is unrelated.

 

[Nelo]

yea i meant 8b, what about 8c? how am i suppposed to know which factors have which exponents?

A degree 6 function with a negetive leading coefficiant. If i was to guess that I would get to... -x (x+6) (x-5)

How am i supposed to know where the exponents go? or do i just add 2s and 3s wherever it makes sense..?

 

[vela]

I think you just to need to let go of the idea that there's a unique answer. You want to find a polynomial, not the polynomial, that satisfies the requirements.

 

[Nelo]

What...? how am I supposed to know then

 

[Nelo]

anyone? 8c) , so then the answer i proposed is correct? the exponent can be anywhere because there are more than 1 solution?

 

[vela]

You haven't posted an answer yet.

 

[Nelo]

-x (x+6) (x-5)

, the square can be anywhere?

 

[vela]

If you're referring to 8b, then yes.