Post on 18-Jan-2016
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
then
So, what is ?
What are the remainders with these? What does that mean?
baba
ba
22
22))(( bababa
ba
ba
33
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
divisorremainder
quotient
3665
Division with a remainder
Here’s how it works for polynomials. becomes
The tricky part to keep straight is what is positive and negative.
A remainder of 0 means we have factors.
2
141323 23
x
xxx 1413232 23 xxxx
Try another
Use the same technique here. 2
1232 23
x
xxx
Try another
This time the divisor is not linear, but the process is the same.
12
1332
23
xx
xxx
Try another
You may find this result familiar. 2
83
x
x
A shortcut
When the divisor is linear, we can use something called synthetic division to find a quick answer.
What was our answer before? How does it compare to these numbers?
2
1232 23
x
xxx
Shortcut again
Try it with this one. Watch for spacers!
2
83
x
x