Transcript Long Division of Polynomials 1

Long Division of
Polynomials
A different way to factor (7.5)
SAT Prep
Quick poll!
1.
SAT Prep
Quick poll!
2.
SAT Prep
Quick poll!
3.
POD
Factor:
x2 – x – 12
Can you use that pattern to factor this one
completely?
x4 – x2 – 12
Pattern review
Difference of squares: a2 – b2 = (a + b)(a - b)
Difference of cubes: a3 – b3 = (a - b)(a2 + ab + b2)
Sum of cubes: a3 + b3 = (a + b)(a2 - ab + b2)
Factoring by grouping (split the middle term)
If the discriminant is a perfect square, then the
quadratic trinomial can be factored.
Factoring and division
If
(a  b)( a  b)  a 2  b 2
then
a2  b2
 ab
ab
So, what is
a3  b3
a b
?
What are the remainders with these?
What does that mean?
Division with a remainder
We can divide polynomials, even if we
get a remainder that isn’t zero. In that
case, we don’t factor, but do
something called long division of
polynomials. It’s a lot like long
division of numbers.
Division with a remainder
First, let’s review the following terms,
and do a simple long division problem.
dividend
divisor
remainder
quotient
5 366
Division with a remainder
Here’s how it works for polynomials.
3x 3  2 x 2  13x  14
x2
becomes
x  2 3x 3  2 x 2  13x  14
The tricky part to keep straight is what is positive
and negative.
A remainder of 0 means we have factors.
Try another
Use the same technique here.
x 3  2 x 2  3x  12
x2
Try another
This time the divisor is not linear, but the
process is the same.
x 3  3x 2  3x  1
x 2  2x  1
Try another
You may find this result familiar.
x3  8
x2
A shortcut
When the divisor is linear, we can use
something called synthetic division to find a
quick answer.
x 3  2 x 2  3x  12
x2
What was our answer before? How does it
compare to these numbers?
Shortcut again
Try it with this one. Watch for spacers!
x3  8
x2