Do Now Find the greatest common factor 72

Do Now: Find the greatest common factor: 72 and 96 Simplify

Greatest Common Factor (GCF) � The Greatest Common Factor (GCF) of two or more numbers is the largest number that can divide into all of the numbers.

GCF � To find the GCF, start by writing out the prime factorization. What factors do the numbers have in common? Circle these then multiply the common factors to get your answer. � Example: find the GCF of 42 and 60.

Find the GCF of each of the following: 1. 144, 36 2. 144 x 2 y 4, 36 x 5 y 3 Factor out the GCF: 3. 144 x 2 y 4 + 36 x 5 y 3 4. 14 x 2 – 7 x + 21

Examples 2. Find the GCF of 12 a 3 b 4 and 3 a 5 b. 3. Find the GCF of 7 x 2 y 2 and 10 xy 3.

Remember this warm up problem? � Multiply: � We 2 x(5 x + 3) can work backwards. What if I gave you the answer 10 x 2 + 6 x, and asked you for the original problem?

We can also “undistribute” our warm up problem � Find the GCF then use the distributive property to factor out the GCF. � What are we really doing? Take out the GCF and then write your “leftovers” on the inside. � Example 13: 10 x 2 + 6 x

We can also find the GCF of a polynomial… � Factor out the GCF of all of the terms. � Example 9: 3 x 3 y – 9 x 2 y 2 � Example 10: 12 x 3 – 8 x 2 + 16 x

Factoring by Grouping � You can try to factor by grouping when your polynomial has four terms.

2. GCF first! Then separate your terms in pairs using 3. Find the 1. 4. Always look for a parenthesis. GCF for each binomial. Your answer will be (GCF’s)(leftovers). ***Your leftovers must match***

Example #1 Always look for a GCF first! Then group your terms in pairs using parenthesis. Find the GCF for each binomial. Your answer will be (GCF’s)(leftovers). Your leftovers must match! Factor: 12 x 3 + 3 x 2 + 20 x + 5

Factor: 3 12 x + 2 3 x + 20 x + 5

Check Answer: (3 x 2 + 5) (4 x + 1) Box or FOIL Method 12 x 3 + 3 x 2 + 20 x + 5

Factor: 5 a 2 x – 4 a 2 c + 15 x 2 – 12 xc

Check Answer: (a 2 + 3) (5 x – 4 c) Box or FOIL Method 5 a 2 x – 4 a 2 c + 15 x 2 – 12 xc

Factor: 4 v 3 – 12 v 2 – 5 v + 15

Check Answer: (4 v 2 – 5) (v – 3) Box or FOIL Method 4 v 3 – 12 v 2 – 5 v + 15

Factor: 21 xy – 12 b 2 + 14 xb – 18 by

Check Answer: Can you factor? What happened?

You Try: 8 x 2 + 8 xy + 2 y 2 + 2 xy

Check Answer: (8 x + 2 y) (x + y) Box or FOIL Method 8 x 2 + 8 xy + 2 y 2 + 2 xy

You Try: 21 k 3 – 84 k 2 + 15 k – 60

Check Answer: (21 k 2 + 15) (k – 4) Box or FOIL Method 21 k 3 – 84 k 2 + 15 k – 60

Error Analysis Factor 20 p 3 + 40 p 2 + 15 p + 30. (20 p 3 + 40 p 2) + (15 p + 30) 20 p 2(p + 2) + 15(p + 2)(20 p 2)

Factoring Standard Form: ax 2 + bx + c

How to factor… �We have to find a number that multiplies to the “c” term whose factors add to the “b” term (x +7)(x + 3) = x 2 + 10 x + 21 7 • 3 = 21 7 + 3 = 10

Let’s do some examples… o What d ve they ha to ly multip to? x 2 + 5 x + 6 What do they have to add to?

o What d ve they ha to ly multip to? x 2 + 7 x + 6 What do they have to add to?

x 2 + 14 x + 45 d 2 – 6 d – 27

x 2 – 3 x - 28 x 2 – 16 x + 48

x 2 + 12 x + 32 x 2 - 25

Do Now:

Factoring when a ≠ 1

Recall Standard form ax 2 + bx + c All the examples we have been doing with factoring, a = 1. However this is not always the case. When there is a coefficient on the x 2 term, and a GCF cannot be taken out, we have to factor a bit differently.

Wrong/Right � Steps: 1. Pull out a GCF of the whole trinomial 2. Write the trinomial as two binomials with the “a” term as the coefficient 3. Multiply the “a” term and the “c” term. 4. See what multiples to “ac” and adds to the “b” term 5. Write in the binomials � This is WRONG 6. We have to make it right by reducing by the GCF, if no GCF, we rewrite � This is the RIGHT answer!

3 y 2 + 14 y + 8 ac = 3*8 = 24 *24 +14 1*24 25 2*12 14 3*8 11 4*6 10 W - ( 3 y + 2 ) ( 3 y + 12 ) R - ( 3 y + 2 ) (y + 4 )

2 y 2 – 7 y - 15