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 x^{4 }so I'm not sure where to start.

f (x) =x^{4}-5x + 3

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 $\displaystyle \frac{m}{n}$ 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 $\displaystyle \pm 1$ and $\displaystyle \pm 3$ as its divisors. It is easy to see that $\displaystyle (-1)^4- 5(-1)+ 3= 9$, $\displaystyle (1)^4- 5(1)+ 3= -1$, $\displaystyle (-3)^4- 5(-3)+ 3= 100$, and $\displaystyle (3)^4- 5(3)+ 3= 69$, none of which are 0 so all roots are irrational numbers.

You **can**, for example, determine that $\displaystyle (0)^4- 5(0)+ 3= 3> 0$ while, as above, the value at 1 is -1< 0. That means there must be a root between 0 and 1. $\displaystyle 2^4- 5(2)+ 3= 9> 0$ 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.

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.

Re: How do I simply something like this?

I expect you have a CAS calculator. Use its solve function to solve the equation $\displaystyle \displaystyle \begin{align*} x^4 - 5x + 3 = 0 \end{align*}$.