CK-12-Pre-Calculus Concepts

2.5. Polynomial Long Division and Synthetic Division http://www.ck12.org

2.5 Polynomial Long Division and Synthetic Division

Here you will learn how to perform long division with polynomials. You will see how synthetic division abbreviates
this process. In addition to mastering this procedure, you will see the how the remainder root theorem and the
rational root theorem operate.
While you may be experienced in factoring, there will always be polynomials that do not readily factor using basic
or advanced techniques. How can you identify the roots of these polynomials?

Watch This

MEDIA

Click image to the left for use the URL below.

URL: http://www.ck12.org/flx/render/embeddedobject/60732

http://www.youtube.com/watch?v=brpNxPAkv1c James Sousa: Dividing Polynomials-Long Division

MEDIA

Click image to the left for use the URL below.

URL: http://www.ck12.org/flx/render/embeddedobject/60734

http://www.youtube.com/watch?v=5dBAdzl2Mns James Sousa: Dividing Polynomials-Synthetic Division

Guidance

There are numerous theorems that point out relationships between polynomials and their factors. For example
there is a theorem that a polynomial of degreenmust have exactlynsolutions/factors that may or may not be real
numbers. TheRational Root Theoremand theRemainder Theoremare two theorems that are particularly useful
starting places when manipulating polynomials.

• TheRational Root Theoremstates that in a polynomial, every rational solution can be written as a reduced
  fraction

(

x=pq

)

, wherepis an integer factor of the constant term andqis an integer factor of the leading

coefficient. Example A shows how all the possible rational solutions can be listed using the Rational Root

Theorem.

• TheRemainder Theoremstates that the remainder of a polynomialf(x)divided by a linear divisor(x−a)
  is equal tof(a). The Remainder Theorem is only useful after you have performed polynomial long division
  because you are usually never given the divisor and the remainder to start. The main purpose of the Remainder
  Theorem in this setting is a means of double checking your application of polynomial long division. Example
  B shows how the Remainder Theorem is used.