# How do I simply something like this?

• Jul 12th 2013, 12:42 PM
Urpes
How do I simply something like this?
The question says: Graph and describe - (in the description i must include the roots, and their multiplicity) the easiest way I can think of finding the roots is to simplify this, however I've never simplified an function with the term x4 ​so I'm not sure where to start.
f (x) =x4-5x + 3
• Jul 12th 2013, 01:00 PM
HallsofIvy
Re: How do I simply something like this?
I'm not sure what you mean by "simplify". That's about as simple as it is going to get. And, as for actually finding the roots, I doubt you will be able to do that. There is a "quartic equation" but it is very complicated.

You can, for example, use the "rational root theorem" that says that if $\frac{m}{n}$ is a root of a polynomial with integer coefficients then n must divide the leading coefficient and m must divide the constant term. Here the leading coefficient is 1 and the constant term is 3 which has only $\pm 1$ and $\pm 3$ as its divisors. It is easy to see that $(-1)^4- 5(-1)+ 3= 9$, $(1)^4- 5(1)+ 3= -1$, $(-3)^4- 5(-3)+ 3= 100$, and $(3)^4- 5(3)+ 3= 69$, none of which are 0 so all roots are irrational numbers.

You can, for example, determine that $(0)^4- 5(0)+ 3= 3> 0$ while, as above, the value at 1 is -1< 0. That means there must be a root between 0 and 1. $2^4- 5(2)+ 3= 9> 0$ so there is another root between 1 and 2. I believe, though I can't prove it, that the other two roots are not real numbers.
• Jul 12th 2013, 01:41 PM
Urpes
Re: How do I simply something like this?
Thanks for the response. Im in grade 12 applied math so I don't expect thats what they want me to do. I guess I'm supposed to use the graphing software provided for me to just identify the roots by seeing them, it just seemed a little too simple. Wow that was some confusing stuff there you just did.
• Jul 12th 2013, 11:57 PM
Prove It
Re: How do I simply something like this?
I expect you have a CAS calculator. Use its solve function to solve the equation \displaystyle \begin{align*} x^4 - 5x + 3 = 0 \end{align*}.