Chapter 2.3: Real Zeros of Polynomial Functions

There are Two Types of ways to divide Polynomial Functions:

• Long Division:

This method usually takes a lot longer and fills up a lot of space. Most people prefer to use the second method when possible. However, in all cases long division will work when Synthetic will not.

• Synthetic Division:

Rules for Synthetic Division: 

• To use Synthetic Division, the factor that is dividing the polynomial must be to the factor of 1, no larger, no smaller.
• The remainder must be added to the end of the answer above the factor used for dividing

ex.

Remainder Theorem: 

• When using Synthetic Division, if you plug in a number that results in a remainder, the number and the reminder is a point on the graph. of the polynomial being divided.
• For the equation, , when the number, 1 or (x-1), is used in synthetic division, a remainder of 1 is left over. We can then say that the point (1,1) is on the graph of.

Rational Zero Test:

• In a polynomial, take the constant and divide it by the leading coefficient to find all the possible x-intercepts.
• For the equation,, all zeros can be found by taking every number from the following list.. and putting them through synthetic division to see which ones produce a remainder of zero. 
• If the numbers in a equation add up to zero then the number, 1, will work.