1. 6. 7 Dividing a Polynomial by a Monomial 6
6.7 Dividing a Polynomial by a Monomial 6.8 Dividing a Polynomial by a Polynomial 7.1 The Greatest Common Factor, Factoring by Grouping

2. When dividing a polynomial by a monomial you will divide every single term in the polynomial by the monomial.
Remember to use your rules for exponents when dividing like bases.

6. 6.8 Dividing a Polynomial by a Polynomial
This is much more difficult and we will use a long division technique (but really as long as we can multiply and add polynomials we will then effectively divide them)

7. Examples

9. YOU NEED PLACE HOLDERS

11. 7.1 The Greatest Common Factor, Factoring by Grouping

12. When we find products we are multiplying together several terms
When we find products we are multiplying together several terms. Those terms are referred to as the FACTORS for that PRODUCT.
3 x 15 = and 15 are the factors
(x+2)(x+3) = x2+5x+6 .. (x+2) and (x+3) are the factors

13. Greatest Common Factor
The greatest common factor for a polynomial is the largest monomial that divides (is a factor of) each term of the polynomial.
The largest monomial is the monomial with the largest coefficient and largest power that is a common factor for each term in the polynomial.

14. Find the greatest common factor for…
Look at the coefficients first, then the variable

15. Find the greatest common factor for…
Look at the coefficients first, then the variable

16. When asked to factor the greatest common factor from a quantity, we are reversing the distributive property. So when you are done you should be able to distribute and get back to what you started with.
Factor the greatest common factor from
5x+15

17. Factor the greatest common factor from

18. Factor the greatest common factor from

19. Factor the greatest common factor from

20. Factor by grouping
At times we may have a group of terms in which there is not a greatest common factor other than 1 amongst the group. But we still want to factor. Our next strategy will be to group the terms in pairs and focus on factoring something out of each group.

21. Example

22. Example

23. Example

24. Example

25. Example same problem but alternative approach

26. As we saw in the last example the grouping really didn’t matter we still achieved the same result. This is a consistent occurance. Sometimes you may have to re-write the polynomial and re-arrange terms to recognize how terms should be grouped.