This document discusses dividing polynomials by binomials. It provides examples of polynomials that can and cannot be divided by binomials. Specifically, it shows that x^2 + 5x + 7 cannot be factored, but can be divided by (x+2), yielding an "ugly" quotient of (x+3) + 1/(x+2). It also notes that complicated polynomials that are hard to simplify can still be divided by binomials, again yielding an ugly quotient. Further examples provided include dividing 4x^3 - 1 + 8x by 4 + 4x and 6y^3 - 4y^2 - 9y - 3 by 2y^2 - 3. The document concludes with assigning exercises 1

1. Dividing Polynomials by Binomials 2
Sometimes factoring polynomials
is difficult or impossible.
Can these polynomials be divided
by binomials?

2. can you factor x 2 + 5x + 7?
NO!
(x + 3) (x + 2) + 1
Not factors!

3. 2 + 5x + 7) by (x+2)?
So can you divide (x
YES!
....but it's ugly.
(x+3)(x+2) 1
(x + 2) + (x + 2) =
(x+3) + 1
(x + 2)

4. What if it is a really complicated
polynomial that is really hard to simplify?
Can it be divided?
YES!
....but it's ugly

5. x 3 + 3x2 + 3x + 1 (x+1)
2
x + 2x + 1
x+1 3 + 3x2 + 3x + 1
x
3 + x2 )
- (x
2x 2 + 3x
- (2x 2 + 2x)
x+1
- (x + 1)
0

6. 2 + 5x + 7 (x + 2)
x
(x + 2) x 2 + 5x + 7

8. 4x 3 - 1 + 8x 4 + 4x
3 + 0x2 + 8x - 1
4x + 4 4x

9. 3 - 4y2 - 9y - 3 2y2-3
6y
a3 + 5a - 18 a -2

10. Assignment
exercise 8 1 - 4 and 8
exercise 9 1- 8