Thread: Consider the function p(x) = ax^5 +bx^4+cx^3+dx^2 + mx +n

1. Consider the function p(x) = ax^5 +bx^4+cx^3+dx^2 + mx +n

Consider the function p(x) = ax^5 +bx^4+cx^3+dx^2 + mx +n. Assume that a>0 and that the polynomial has zeros of r(multiplicity 1), s (multiplicity 2), and t(multiplicity 2) where r<s<t.
a) Determine any intervals where P(x)>0
b) Determine any intervals where P(x)<0
c) Determine the number of local maximum points of P(x)
d) Determine the number of local minimum points of P(x)
e) Sketch a possible graph of P(x)

I don't get this question one bit, can someone set me on the path or solve a few of those like a) and d), so I understand them. Plus, just give me a few pointers on how I could start e).
Thanks!

2. Re: Consider the function p(x) = ax^5 +bx^4+cx^3+dx^2 + mx +n

Ok so "the polynomial has zeros of r(multiplicity 1), s (multiplicity 2), and t(multiplicity 2) where r<s<t" means that $(x-r)$ and $(x-s)^2$ and $(x-t)^2$ are factors. The multiplicity refers to the exponents of the factors.

Since the polynomial is of degree 5 (ie highest power of $ax^5 +bx^4+cx^3+dx^2 + mx +n$) and you've accounted for 1 +2+2=5 factors,
$ax^5 +bx^4+cx^3+dx^2 + mx +n = a(x-r)(x-s)^2(x-t)^2$.
Can you imagine if you expanded out the right hand side the leading term would be $ax^5$?

So far so good?

3. Re: Consider the function p(x) = ax^5 +bx^4+cx^3+dx^2 + mx +n

I would start this question by doing part (e) first. You have an (odd) 5th degree polynomial ... you should already know its end behavior from the fact that it is odd-degree and has a positive leading coefficient. Second, you need to know about the behavior of the polynomial's graph at roots of different multiplicities. Check out the link for a decent explanation for both topics.

polynomials.html

Based on the information given and the above ideas, I was able to make a rough sketch of a graph with adequate info to answer parts (a) thru (d) ...

4. Re: Consider the function p(x) = ax^5 +bx^4+cx^3+dx^2 + mx +n

Originally Posted by skeeter
I would start this question by doing part (e) first. You have an (odd) 5th degree polynomial ... you should already know its end behavior from the fact that it is odd-degree and has a positive leading coefficient. Second, you need to know about the behavior of the polynomial's graph at roots of different multiplicities. Check out the link for a decent explanation for both topics.

polynomials.html

Based on the information given and the above ideas, I was able to make a rough sketch of a graph with adequate info to answer parts (a) thru (d) ...
Oh yes, doing e) first makes it so much more easier to understand.
So for a) the answer is any x values greater than r is where p(x) is positive, right? For b) the answer is any x values less than r is where p(x) is negative, right?

For c) the answer is 2 and d) the answer is 2. Am I correct?

Just one last question. Between r and s on the graph, how did you know that the y value is bigger than the y value between s and t?

5. Re: Consider the function p(x) = ax^5 +bx^4+cx^3+dx^2 + mx +n

Originally Posted by Latinized
Oh yes, doing e) first makes it so much more easier to understand.
So for a) the answer is any x values greater than r is where p(x) is positive, right? Right, except for x= s and x=t, because then P(x) =0 not >0
For b) the answer is any x values less than r is where p(x) is negative, right? Yep

For c) the answer is 2 and d) the answer is 2. Am I correct? Yep. Yep.

Just one last question. Between r and s on the graph, how did you know that the y value is bigger than the y value between s and t?
It may not be, you don't know how high the max points are. That's why (e) says to sketch a possible graph.
.

6. Re: Consider the function p(x) = ax^5 +bx^4+cx^3+dx^2 + mx +n

Just one last question. Between r and s on the graph, how did you know that the y value is bigger than the y value between s and t?
If I had made them the same height, would you have assumed both relative maxima were equal in value?