Synthetic Division
Grade Level(s): 9-12
Subject(s):
Description: Synthetic division
is an easy method for dividing polynomials by a linear factor.
Objectives: Given a polynomial
of second degree or higher, students will be able to determine all linear factors that divide evenly into the polynomial.
Prerequisties: Previous
knowledge of polynomials and factors.
Concepts: Synthetic division
is called “synthetic” because it provides a comfortable alternative to long division of polynomials. However,
it only works for linear factors. The method uses the coefficients of each term in the polynomial to find possible linear
factors that divide evenly into the polynomial.
For example, let’s take the polynomial x4 + 2x3 – x2 + 7x – 9 = 0. Let’s make the synthetic division
chart:
Then you put possible factors in the lower left-hand corner, like for
example (x – 1), which means that x
= 1, so we put 1 in the box (not -1).
The first coefficient is dropped down. Then it is added to the following
one. This sum is added to the following one, and so on. In order for (x –
1) to be a linear factor, the last sum has to yield zero.
|
|
1 |
2 |
-1 |
7 |
-9 |
|
|
↓ |
1 |
3 |
2 |
9 |
|
1 |
1 |
3 |
2 |
9 |
0 |
If the last sum is not 0, then this sum is a remainder over the original
polynomial. The student can also try a different factor to see if it is a linear factor of the original polynomial.
Once the original polynomial is divided by the linear factor, the numbers
in the bottom row are the coefficients of the remaining polynomial. This means that when x4
+ 2x3 – x2
+ 7x – 9 is divided by (x –
1), we are left with x3 + 3x2
+ 2x + 9, or
(x – 1)( x3 + 3x2 + 2x
+ 9) = x4 + 2x3
– x2 + 7x – 9
The method can be repeated to find with the new third degree polynomial
to find another linear factor until all linear factors are found.
Using synthetic division, students will be more likely to remember how
to factor polynomials, and as a result, show a better understanding of the subject.
Assessment: Students have
to use first long division of polynomials and then synthetic division on a given polynomial to compare the methods.
Lesson Plan Layout taken from:
http://eduref.org/cgi-bin/printlessons.cgi/Virtual/Lessons/Mathematics/Algebra/ALG0002.html
