Question 1051320
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The first thing you need is that complex zeros always appear in conjugate pairs.  So write the given zero in complex number, *[tex \Large a\ +\ bi] form:  *[tex \Large 0\ +\ (-i)\ =\ 0\ -\ i].  The conjugate of *[tex \Large a\ +\ bi] is *[tex \Large a\ -\ bi], so the companion zero to the one given is *[tex \Large 0\ -\ (-i)\ =\ 0\ +\ i]


If *[tex \Large \alpha] is a zero of a polynomial, then *[tex \Large x\ -\ \alpha] must be a factor of the polynomial. Hence, we now know two of the factors of the given polynomial, to wit:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \  x\ -\ (0\ -\ i)\ =\ x\ +\ i]


and


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ x\ -\ (0\ +\ i)\ =\ x\ -\ i]


Since this is a binomial conjugate pair, their product is the difference of two squares.  Remember that *[tex \Large i^2\ =\ -1].  Hence:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \  (x\ +\ i)(x\ -\ i)\ =\ x^2\ -\ (-1)\ =\ x^2\ +\ 1]


Use polynomial long division to divide the original polynomial function by *[tex \Large x^2\ +\ 1]


<pre>
                                   x  +  11
              -----------------------------
x^2 + 0x + 1  |  x^3  +  11x^2  +  x  +  11
                 x^3      0x^2  +  x
                 --------------------------
                         11x^2  + 0x  +  11
                         11x^2  + 0x  +  11
                        -------------------
                                          0
</pre>

So the third and final factor is *[tex \Large x\ +\ 11] and therefore the third and final zero is -11


By the way, saying x2 or x3 to mean x squared or x cubed is confusing. Use the caret mark (^) to indicate raising to a power, such as x^5 or e^x, which we all understand to mean *[tex \Large x^5] or *[tex \Large e^x].  If you want more information on rendering mathematical expressions in plain text, review <a href="http://www.purplemath.com/modules/mathtext.htm">Formatting Math as Text</a> (Note that there are four pages of information).


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 \ \ \ \ \ \ \ \ \ \  

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