SOLUTION: Use the remainder theorem to determine whether x - 2 is a factor of f(x) = x^3 + 3x^2 - x - 18

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Question 1084384: Use the remainder theorem to determine whether x - 2 is a factor of
f(x) = x^3 + 3x^2 - x - 18
Answer by ikleyn(52781)   (Show Source): You can put this solution on YOUR website!
.
The Remainder Theorem says: 


     The binomial    divides the polynomial    if and only if the value of    is the root of the polynomial  ,  i.e.  .


So,  to check if  (x-2)  is the factor of  f(x) = x^3 + 3x^2 - x - 18,  we need to calculate the value  f(2).

It is  f(2) = = 8 + 3*4 - 2 - 18 = 0.

Thus according to the Remainder Theorem  (x-2)  is the factor of the given polynomial  f(x).



Solved.


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   Theorem   (the remainder theorem)
   1. The remainder of division the polynomial    by the binomial    is equal to the value    of the polynomial.
   2. The binomial    divides the polynomial    if and only if the value of    is the root of the polynomial  ,  i.e.  .
   3. The binomial    factors the polynomial    if and only if the value of    is the root of the polynomial  ,  i.e.  .


See the lesson
    - Divisibility of polynomial f(x) by binomial x-a
in this site.


Also,  you have this free of charge online textbook in ALGEBRA-II in this site
    ALGEBRA-II - YOUR ONLINE TEXTBOOK.

The referred lessons are the part of this online textbook under the topic
"Divisibility of polynomial f(x) by binomial (x-a). The Remainder theorem".


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