Question 240748
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There are two ways to tell if a given number is a zero of a given polynomial function.


First, and probably easiest conceptually, is:


*[tex \LARGE a] is a zero of the polynomial function *[tex \LARGE f(x)] if and only if *[tex \LARGE f(a)\ =\ 0]


For one of your examples:


*[tex \LARGE 4] is a zero of the polynomial function *[tex \LARGE f(x)\ =\ x^3\ -\ 6x^2\ -\ 27x\ +\ 140] if and only if *[tex \LARGE f(4)\ =\ (4)^3\ -\ 6(4)^2\ -\ 27(4)\ +\ 140\ =\ 0].


You can do your own arithmetic to check whether 4 is or is not a zero.


The second method is by using Polynomial Long Division.  The Polynomial Long Division method relies on the following theorem:


*[tex \LARGE a] is a zero of the polynomial function *[tex \LARGE f(x)] if and only if *[tex \LARGE x\ -\ a] is a factor of *[tex \LARGE f(x)].  That is to say that there exists a polynomial function *[tex \LARGE p(x)] such that *[tex \LARGE f(x)\ =\ p(x)(x\ -\ a)]


This leads to the process of polynomial long division.  For your other example, divide *[tex \LARGE x^3\ -\ 6x^2\ -\ 27x\ +\ 140] by *[tex \LARGE x\ -\ (-5)\ =\ x\ +\ 5] using polynomial long division.  If the result is a quotient with a remainder of zero, then *[tex \LARGE x\ +\ 5] is a factor meaning -5 is a zero.  If there is a non-zero remainder (meaning p(x) in the theorem above is a rational function rather than a polynomial function), then *[tex \LARGE x\ +\ 5] is NOT a factor meaning -5 is NOT a zero. 


Use the procedure for polynomial long division found at either of the following websites:


http://www.purplemath.com/modules/polydiv2.htm


or


http://en.wikipedia.org/wiki/Polynomial_long_division


John
*[tex \LARGE e^{i\pi} + 1 = 0]
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