Download 3.3 Polynomial Division: Factors and Zeros

3.3 Polynomial Division: Factors and Zeros
In this section, we factor polynomials and find their zeros.
Polynomial Long Division
If the result of dividing a polynomial
, then the following equations hold:
by a divisor
polynomial
is a the quotient
divisor quotient
with a remainder
remainder
Synthetic Division
To divide
by
, construct the following table:
|
_____________________________
Be sure to write for missing coefficients . Beginning from the left, add each column and then
write the product of the result and above the line in the next column. The final number written is
the remainder. The other numbers are the coefficients of the quotient.
The Remainder/Factor Theorem
Let
be a polynomial function. Then:
1.
2.
equals the remainder
if and only if
when
is divided by
is a factor of
Factoring a Polynomial (and Finding Its Zeros)
If our usual factoring methods (for quadratics or some special cubics) don’t work, then the
following method is the best we can do (with a few improvements in the next section):
1.
2.
3.
4.
Find a zero of the polynomial
Use synthetic division to divide
by
. The remainder should be
Write
, where
is the quotient of the division.
Repeat the process to factor
further if necessary.
.