Question

True or false.

Answer 1

The number 3 is an upper bound for the set of roots of this polynomial function.

Answer 2

\[f(x) = 3x^4 - 5x^3 - 5x^2 + 5x + 2\]

Answer 3

I used synthetic division and all of the numbers were positive, so I thought it was an upper bound.

But then I noticed that 3 wasn't even a root possibility (not a factor of the constant, 2). So can 3 even be an upper bound?

Answer 4

@mathmate @mathstudent55 @agent0smith

Answer 5

"I used synthetic division and all of the numbers were positive, so I thought it was an upper bound."

That sounds right. You can check by graphing - if there no roots beyond x=3, it is an upper bound.

I don't think it matters if it's a *possible* root or not.

Answer 6

Whew! @agent0smith You were correct. Thanks :)

Answer 7

there is a difference between upper bound and least upper bound.
You are right that there is probably an upper bound less than 3, but 3 still is an upper bound i.e. greater than the roots.

Answer 8

Thanks, I appreciate the help. I was just confused because 3 wasn't a possible root... but it turned out to be right.

Answer 9

https://www.wolframalpha.com/input/?i=f(x)+%3D+3x%5E4+-+5x%5E3+-+5x%5E2+%2B+5x+%2B+2

See, clearly no roots beyond 3

And yes phi probably has a point with the least upper bound.

Answer 10

If we use the Descartes rule of signs, we know there can only be a maximum of two real positive roots.
The factor theorem says that, if the roots are rational (which is not given), then the possible roots, are, from maximum to minimum, 2,1,.... which means that the upper bound is 2+1+(other negative roots), or the upper bound is 3.

Note: we \(know\) that the product of the roots is 2/3, and sum of the roots is -5/3 (using the last two coefficients after converting to a monic polynomial), so the "upper bound" is superfluous! lol