Question 797714
<font face="Times New Roman" size="+2">


In the first place you cannot find "the" zero of a 4th degree polynomial function.  That is because, according to the Fundamental Theorem of Algebra, any 4th degree polynomial function has 4 zeros.  In the second place, you don't have a function, you have an equation.


*[tex \LARGE \ \ \ \ \ \ \ \ \ \  f(x)\ =\ x^4\ -\ 14x^3\ +\ 56x^2\ -\ 98x\ +\ 343]


is a quartic or 4th degree function.  Functions have zeros.  nth degree functions have n zeros, counting all multiplicities.


*[tex \LARGE \ \ \ \ \ \ \ \ \ \  x^4\ -\ 14x^3\ +\ 56x^2\ -\ 98x\ +\ 343\ =\ 0]


is a quartic equation.  Equations have roots, one for each degree of the polynomial.



Use the Rational Roots Theorem.  Since the lead coefficient is 1 and 343 is 7 cubed, the only possible rational roots are 7 and -7.


Test with Synthetic Division:
<pre>
 7  |   1 -14  56 -98  343
    |       7 -49  49 -343
--------------------------
        1  -7   7 -49    0
</pre>
That worked right off the bat, so now we know that we have


*[tex \LARGE \ \ \ \ \ \ \ \ \ \  f(x)\ =\ (x\ -\ 7)\left(x^3\ -\ 7x^2\ +\ 7x\ -49\right)]


Since you don't want to argue with success, let's try the same thing again.
<pre>
 7  |   1 -7  7 -49
    |      7  0  49
--------------------------
        1  0  7   0
</pre>
Success again!  So now we know:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ f(x)\ =\ (x\ -\ 7)^2(x^2\ +\ 7)]


And since the final factor is a quadratic, it can be solved by any convenient means.  If you are only concerned with real number zeros, you can quit now because there are no real roots for *[tex \Large x^2\ +\ a\ =\ 0] for any positive value of *[tex \Large a].  However, if you need the complex zeros also, re-write the quadratic factor as an equation thus:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \  x^2\ -\ (-7)\ =\ 0]


And factor as the difference of two squares:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \  \left(x\ +\ i\sqrt{7}\right)\left(x\ -\ i\sqrt{7}\right)]


In summary, including the complex roots:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \  x\ =\ 7]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \  x\ =\ 7]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \  x\ =\ i\sqrt{7}]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \  x\ =\ -i\sqrt{7}]


For a total of 4 zeros, as advertised.


John
*[tex \LARGE e^{i\pi}\ +\ 1\ =\ 0]
<font face="Math1" size="+2">Egw to Beta kai to Sigma</font>
My calculator said it, I believe it, that settles it
<div style="text-align:center"><a href="http://outcampaign.org/" target="_blank"><img src="http://cdn.cloudfiles.mosso.com/c116811/scarlet_A.png" border="0" alt="The Out Campaign: Scarlet Letter of Atheism" width="143" height="122" /></a></div>
</font>