Thread: Find a bound on the real zeros of the polynomial function

I have a question if someone has a few minutes to explain it to me.

I am going over my practice test and one of the questions if the thread title. I am unsure if I understand what it is asking. Does it want what the bounds are that the real zeros will be in?

The function is x^5+2x^4+2x^3-7x^2+x+4

According to the test, the answer is -8 and 8 but I have no idea how to get there. Looking at the graph of the function, I would think it would fall between -1 and 1, but again, I am unsure as to what it is asking for when asking for the bounds.

Our instructor missed class the other day so we have not went over this and I am trying to stay ahead.

Originally Posted by wevie

I am unsure if I understand what it is asking. Does it want what the bounds are that the real zeros will be in?

Yes, it asks to find a positive constant C such that for any real x, if x is a root of the polynomial, then |x| <= C.

There are several ways to find the bounds on polynomial roots, but it seems that Cauchy’s bound (see here and here) is sufficient for this example. However, there is no substitute to finding out what was covered in your course.

Originally Posted by emakarov

Yes, it asks to find a positive constant C such that for any real x, if x is a root of the polynomial, then |x| <= C.

There are several ways to find the bounds on polynomial roots, but it seems that Cauchy’s bound (see here and here) is sufficient for this example. However, there is no substitute to finding out what was covered in your course.

This is what I have in my textbook about bounds. A positive number M is a bound on the zeros of a polynomial if every zero r lies between -M and M, inclusive. That is, M is a bound to the zeros of a polynomial f is -M less than/equal to any zero of f less than/equal to M.

In looking at the textbook, I think I need to take the leading coefficient which is 1 and add it to the absolute value of -7 which is 7 and that is 8. Or I add the absolute value of all the coefficients and take which ever is the lesser number?

Could someone please explain to me or show me how the answer is -8 and 8?

Like I said, my instructor missed class last week and while we will still cover what we missed that day, I am sure she will rush through it since this is an accelerated summer course. I am simply trying to make some sense of what we will cover in class before we cover it.

Have you looked at the links I provided? Cauchy bound is exactly 8 for this polynomial.

Originally Posted by wevie

In looking at the textbook, I think I need to take the leading coefficient which is 1 and add it to the absolute value of -7 which is 7 and that is 8. Or I add the absolute value of all the coefficients and take which ever is the lesser number?

Which formula for the bound does your textbook have? I hope you are not just coming up with formulas out of the blue.

Originally Posted by wevie

Like I said, my instructor missed class last week and while we will still cover what we missed that day, I am sure she will rush through it since this is an accelerated summer course. I am simply trying to make some sense of what we will cover in class before we cover it.

There are many methods for finding bounds on polynomial roots, and it is difficult to predict what your instructor decided to include in the course.

Originally Posted by emakarov

Have you looked at the links I provided? Cauchy bound is exactly 8 for this polynomial.

Which formula for the bound does your textbook have? I hope you are not just coming up with formulas out of the blue.

There are many methods for finding bounds on polynomial roots, and it is difficult to predict what your instructor decided to include in the course.

Yes. I looked at the link and while it looks similar to what is in the textbook, it also seems much more complicated. I think I have it figured out though and class is in a couple of hours so hopefully that will help me make more sense of it.