Question 804181
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Properties of Polynomials:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ a_nx^n\ +\ a_{n-1}x^{n-1}\ +\ \... \ +\ a_2x^2\ +\ a_1x\ +\ a_0]


The Fundamental Theorem of Algebra guarantees that an *[tex \Large n]-th degree polynomial function will have *[tex \Large n] zeros, counting multiplicities, some or all of which may be complex roots.


Complex roots always appear in conjugate pairs, so:


If *[tex \Large n] is even, the polynomial may have 0, 2, 4, ... , n real number zeros.


If *[tex \Large n] is odd, the polynomial is guaranteed to have at least one real number zero, but may have 1, 3, 5, ..., n real number zeros.


<i><b>Rational Roots Theorem</b></i>:  If a polynomial has a rational number zero, then it will be of the form *[tex \Large \pm\frac{p}{q}] where *[tex \Large p] is an integer factor of the constant cooefficient, *[tex \Large a_0], and *[tex \Large q] is an integer factor of the lead coefficient, *[tex \Large a_n]


<i><b>Corollary to The Fundamental Theorem of Algebra</b></i>:  If *[tex \Large \alpha] is a zero of a polynomial, then *[tex \Large (x\ -\ \alpha)] is a factor of the polynomial.


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*[tex \LARGE \ \ \ \ \ \ \ \ \ \ f(x)\ =\ x^3\ -\ x^2\ -\ 37x\ -\ 35]


is a third degree polynomial function, hence there is either 1 real zero or there are 3 real zeros.  According to the Rational Roots Theorem, if any of these zeros are rational numbers, they are *[tex \Large \pm1], *[tex \Large \pm5], or *[tex \Large \pm7]


We test using Synthetic Division. First test 1:
<pre>
1  |  1   -1   -37   -35
   |       1     0   -37
   ----------------------
      1    0   -37   -72  Last result non-zero, 1 is not a zero.

Try -1

-1 |  1   -1   -37   -35
   |      -1     2    35
   ----------------------
      1   -2   -35     0  Last result is zero, -1 is a zero
</pre> 
 
Hence *[tex \Large (x\ +\ 1)] is a factor and *[tex \Large (x^2\ -\ 2x\ -\ 35)] is the other factor.  But  *[tex \Large (x^2\ -\ 2x\ -\ 35)] is a factorable quadratic.


I'll leave it to you to finish this.


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
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