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) About Me  (Show Source):
You can put this solution on YOUR website!
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The Remainder Theorem says: 


     The binomial  x-a  divides the polynomial  f%28x%29  if and only if the value of  a  is the root of the polynomial  f%28x%29,  i.e.  f%28a%29+=+0.


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) = 2%5E3+%2B+3%2A2%5E2+-+x+-+18 = 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  f%28x%29  by the binomial  x-a  is equal to the value  f%28a%29  of the polynomial.
   2. The binomial  x-a  divides the polynomial  f%28x%29  if and only if the value of  a  is the root of the polynomial  f%28x%29,  i.e.  f%28a%29+=+0.
   3. The binomial  x-a  factors the polynomial  f%28x%29  if and only if the value of  a  is the root of the polynomial  f%28x%29,  i.e.  f%28a%29+=+0.


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".