Applying the Remainder Theorem

Date: 11/19/2005 at 18:13:33
From: Craig
Subject: Remainder Theorem

Given f(x) = 7x^4 + 9x^3 + 4x^2 - 4x + 16, use the Remainder Theoreom
to find f(2).  I don't understand the theorem as a whole.

Date: 11/20/2005 at 00:09:37
From: Doctor Wilko
Subject: Re: Remainder Theorem

Hi Craig,

Thanks for writing to Dr. Math!  The remainder theorem says that when
a polynomial f(x) is divided by (x-r), the remainder is f(r).  In
other words, the remainder of the division process has the same value
you would get by plugging r into the function and evaluating it.

So the remainder theorem, in your case, says that when the polynomial 

  f(x) = 7x^4 + 9x^3 + 4x^2 - 4x + 16

is divided by 

  (x - 2),

the remainder is the same as what you get by evaluating f(2).  Let's
see how to use this.  Your question is something like,

 "Use the remainder theorem to evaluate f(2)."

You're supposed to do the following:

Use division (synthetic or long) to divide the polynomial f(x) by  
(x-2) and note the remainder of 208.  I recommend synthetic:

  2 | 7   9   4   -4   16
         14  46  100  192
     --------------------
      7  23  50   96  208

From this (because of the theorem), you can state that f(2) = 208 
(without actually calculating f(2) directly).   

Why is this useful?  

f(2) is somewhat complicated to calculate by hand with all the 
exponents, but I can get the answer to f(2) by doing a simpler problem 
(division of the polynomial f(x) by (x-2)) and taking note of the 
remainder.  This remainder is the answer to f(2).  The remainder 
theorem gives me an alternate way to calculate f(2).

--------------------------------
This also leads into another theorem, the factor theorem.

Basically the factor theorem says that if you take a polynomial f(x) 
and calculate f(r), and f(r) = 0, then (x-r) is a factor of f(x).  

To relate this to your question, is f(2) = 0?

No.  We found that f(2) = 208.  Therefore (x-2) is not a factor of   
f(x) = 7x^4 + 9x^3 + 4x^2 - 4x + 16.  This means that (x-2) doesn't 
divide evenly into the polynomial f(x).

To summarize this, the factor theorem gives you a quick way to see if 
some (x-r) is a factor of a polynomial f(x).  Just calculate f(r).  If 
f(r) = 0, then (x-r) is a factor of f(x).

---------------------------------
If you look back at these two theorems, you'll see that they are 
closely related.

Here's a couple more links on these topics from our archives:

  Applying the Remainder Theorem
    http://mathforum.org/library/drmath/view/53203.html 

  Factoring and the Factor Theorem
    http://mathforum.org/library/drmath/view/53146.html 

Does this help?  Please write back if you have further questions.
:-)


- Doctor Wilko, The Math Forum
  http://mathforum.org/dr.math/