Remainder, Rational and Factor Theorems

The Remainder Theorem

The Remainder Theorem is basically a method to evaluate a polynomial at a given value.
For example:

polynomial p(x) = 5x⁴ + 2x³ + 4x

If we want to evaluate this function at x = 2, we can do two things.
One is the simple plug the value 2 in and evaluate, but the Remainder Theorem can also work.
First step in using the Remainder Theorem is synthetic division. 
So for this example and x = 2 we have:

   2  |  5   2    0     4    0
       |      10   24  48  104
          5  12   24  52  104

The remainder is 104.  This should match what we would get if we plugged x = 2 into the original polynomial.
Lets check:

5 * 16 + 2 * 8 + 4 * 2 = 104

An important reason to use the Remainder Theorem is evaluate a value of x for the root(s) of a polynomial.
If for a given x the remainder is 0 then x is a root.

The Rational Root Theorem

The Rational Root Theorem states that it is possible to find all possible rational roots of a function by determining:

+          factors of the constant       
      factors of the leading coefficient 

     
For example, consider the equation:

The possible roots would look like

 +    1,3   

        1

meaning that all the possible roots are: 1, -1, 3, -3

 

The Factor Theorem

The factor theorem is essentially the remainder theorem in reverse. If a polynomial is synthetically divided by x=a and a remainder of zero is found then x=a is a zero which is essentially the Remainder Theorem, but what is key to the Factor Theorem is this also means (x-a) is a factor of the polynomial.

Helpful sites:

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

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

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




Jack Kelly