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Math Help - How do I simply something like this?

  1. #1
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    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
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  2. #2
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    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.
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    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.
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  4. #4
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    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*}.
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