1. Long Division Algorithm and Synthetic Division!!!
Sec. 3.3a
Homework: p odd

2. We use a similar process when dividing polynomials!!!
First, let’s work through this…
We can write our answer as:
We use a similar process
when dividing polynomials!!!
Remainder!

3. Division Algorithm for Polynomials
Let f(x) and d(x) be polynomials with the degree of f greater
than or equal to the degree of d, and d(x) = 0. Then there are
unique polynomials q(x) and r(x), called the quotient and
remainder, such that
where either r(x) = 0 or the degree of r is less than the degree
of d. The function f(x) is the dividend, d(x) is the divisor, and
if r(x) = 0, we say d(x) divides evenly into f(x).
Fraction form:

4. Using Polynomial Long Division
Use long division to find the quotient and remainder when
is divided by Write a summary
statement in both polynomial and fraction form.
 Quotient
 Remainder

5. Can we verify these answers graphically???
Using Polynomial Long Division
Use long division to find the quotient and remainder when
is divided by Write a summary
statement in both polynomial and fraction form.
Polynomial Form:
Fraction Form:
Can we verify these answers graphically???

6. Synthetic Division Synthetic Division is a shortcut method for the
division of a polynomial by a linear divisor, x – k.
Notes:
This technique works only when dividing by a
linear polynomial…
It is essentially a “collapsed” version of the long
division we practiced last class…

7. Synthetic Division – Examples:
Evaluate the quotient:
Coefficients of dividend:
Zero of
divisor: 3 2 –3 –5
–12 6 9 12 2 3 4
Remainder
Quotient

8. 1 –2 3 –3 –2 4 –4 2 1 –2 2 –1 –1 Verify Graphically?
Synthetic Division – Examples:
Divide by and write a
summary statement in fraction form. –2 1 –2 3 –3 –2 4 –4 2 1 –2 2 –1 –1
Verify Graphically?

9. Practice
Divide the above function by
Divide the above function by

10. Yes, x – 3 is a factor of the second
Using Our New Theorems
Is the first polynomial and factor of the second?
Yes, x – 3 is a factor of the second
polynomial

11. Some whiteboard problems…

12. “Fundamental Connections” for Polynomial Functions
For a polynomial function f and a real number k, the following
statements are equivalent:
1. x = k is a solution (or root) of the equation f(x) = 0.
2. k is a zero of the function f.
3. k is an x-intercept of the graph of y = f(x).
4. x – k is a factor of f(x).