2.  When dividing two polynomials, we get
a quotient polynomial and a remainder
polynomial.
 Written: Remainder
Divisor
Quotient
 The degree of the remainder must be
less than the degree of the divisor.

3.  Write in the same format as dividing
numbers.
 Remember:

4.  Use a 0 coefficient for missing terms.
 Divide the 1st term of the dividend by the
1st term of the divisor.
 Example: Divide by

5.  Divide

6.  Divide

7.  Used to evaluate polynomials.
 Example: Evaluate
 Write x value on the left and list
coefficients of f(x).
3 2 0 -8 5 -7

8.  Use synthetic substitution to evaluate

9.  Use synthetic substitution to evaluate
 Now use long division to divide

10.  If a polynomial f(x) is divided by x – k,
then the remainder is r = f(k).
 We just noticed the remainder when f(x)
was divided by x – 2 was 15 and also
f(2) = 15, so r =f(k).
 We also noticed the other numbers
resulting from synthetic substitution
were the coefficients of the quotient.
 We can use this to divide polynomials!

11.  Can be used to divide a polynomial by
an expression of the form x – k.
 Example:

12.  Use synthetic division to divide
 by x – 1
 by x + 2

13.  Use synthetic division to divide

14.  A polynomial f(x) has a factor x – k if
and only if f(k) = 0.
 (when r = 0 for f(x) divided by x – k)
For example: Dividing
gives a quotient of with no
remainder.
Therefore:

15.  Factor
 Since f(-3) = 0, we know x – (-3) or
x+3 is a factor of f(x).
 Use synthetic division to find other
factors.

16.  Factor

17.  Factor

18.  Remember: a zero is an x value for
which f(x) = 0
 To find zeros, factor completely, then
set each factor equal to zero and solve.
 Example: One zero of
is x = 2. Find the other zeros.

19.  One zero of
is x = 7. Find the other zeros.