M3D10 Have out: Bellwork: pencil, red pen, highlighter, GP notebook, Factor completely across the rationals. Be sure to look for any greatest common factor first! 1) 2) 3) –4 –3 35 12 +1 5 7 –4 +3 +2 +3 total:

Multiply the following together using a generic rectangle. b) x2 + 2 a + b x c x3 +2x ac bc + 5 +5x2 +10 + d + ad +bd What if you were given the answers above, but you needed to get back to the original problem. How would you “undo” the answer? Try _________! factoring However, diamond problems will not help because we are not working with __________. trinomials

Group factoring 4 _______________ – when polynomials have ___ or more terms, it may be possible to factor these polynomials by __________ the terms in ______ to help with factoring. grouping pairs a(b + c) = ab + ac Recall the distributive property: _________________. This will help us with group factoring. Example #1: Factor Group Step 1: _______ pairs of terms that have a ________ monomial. common Step 2: Factor the _____ from each group. GCF Step 3: Notice that _____ is a common factor. c + d distributive Step 4: Use the ___________ property to factor out ______. c + d

Example #2 – 3: Factor completely using group factoring. 2) 3) Rewrite in descending order

Practice: Factor completely using group factoring. b)

Multiply the following together using a generic rectangle. b) x2 + xy + y2 x2 – xy + y2 x x3 + x2y x +xy2 x3 – x2y +xy2 – y –x2y –xy2 – y3 + y +x2y –xy2 + y3

What patterns do you notice? All the middle terms cancel out. The only terms left are cubes. The previous patterns lead us to the factoring method of _________________________. Sum and Difference of Cubes Sum: Difference:  Notice how the first factor almost matches the sum or difference of cubes.  Notice that the first and last terms of the second factor are squared.  Notice how the signs are different in the first factor and the middle term of the other factor.

Examples: Use the patterns above to factor the following: ( )( ) x + 4 x2 – 4x + 42 ( )( ) y – 3 y2 + 3y + 32

(2x)2 2x + y – 2xy + y2 3a – 4b (3a)2 + 3a●4b + (4b)2 Examples: Use the patterns above to factor the following: c) d) (2x)2 ( )( ) 2x + y – 2xy + y2 ( )( ) 3a – 4b (3a)2 + 3a●4b + (4b)2

Examples: Use the patterns above to factor the following: – 5xy ( )( ) x – 2 x2 + 5y + (5y)2 + 2x ( )( ) + 22

We can prove that the rule always works for the Difference of Cubes. Let a and b be any number, then: zero Add “______” group Use ______ factoring and factor the ______from each group GCF difference of squares Use __________________ to factor ________ a – b Notice that ______ is a common factor. distributive Use the ___________ property to factor out ______. a – b A similar proof can be given for the Sum of Cubes.

After the quiz, work on the rest of the packet. Quiz Time Clear your desk except for a pencil, highlighter, and a NON-GRAPHING calculator! After the quiz, work on the rest of the packet.