Question 1131744
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*[tex \LARGE \ \ \ \ \ \ \ \ \ \ x^3\ -\ 9x^2\ +\ 23x\ -\ 15\ =\ 0]


3 roots.  3 sign changes for *[tex \Large f(x)]: Maximum 3 positive real roots.


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ (-x)^3\ -\ 9(-x)^2\ +\ 23(-x)\ -\ 15\ =\ 0]


Zero sign changes for *[tex \Large f(-x)]:  Maximum zero negative real roots.


The lead coefficient is 1, the constant term is -15.  Possible rational roots *[tex \LARGE \ \ \ \ \ \ \ \ \ \ \pm1,\ \pm3,\ \pm5,\ \pm15]


but the negative values are excluded by the Rule of Signs, so the possible rational roots are *[tex \LARGE 1,\ 3,\ 5, and\ 15]


Synthetic Division
<pre>

5  |  1   -9   23  -15								
           5  -20   15   
   -------------------
      1   -4    3    0
</pre>


*[tex \LARGE \ \ \ \ \ \ \ \ \ \  (x\ -\ 5)(x^2\ -\ 4x\ +\ 3)]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \  (x\ -\ 5)(x\ -\ 1)(x\ -\ 3)]


Zeros are 1, 3, and 5

								
John
*[tex \LARGE e^{i\pi}\ +\ 1\ =\ 0]
My calculator said it, I believe it, that settles it
<img src="http://c0rk.blogs.com/gr0undzer0/darwin-fish.jpg">
*[tex \Large \ \
*[tex \LARGE \ \ \ \ \ \ \ \ \ \  
								
{{n}\choose{r}}
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