1. Day 136 – Common Factors

2. Factor each polynomial. a)𝟓𝒂𝒎−𝟓𝒂𝒏 b)𝟓 𝒙 𝟑 −𝟑 𝒚 𝟐 c)𝟐 𝒄 𝟒 −𝟒 𝒄 𝟑 +𝟔 𝒄 𝟐
Example 1
Factor each polynomial. a)𝟓𝒂𝒎−𝟓𝒂𝒏 b)𝟓 𝒙 𝟑 −𝟑 𝒚 𝟐 c)𝟐 𝒄 𝟒 −𝟒 𝒄 𝟑 +𝟔 𝒄 𝟐

3. Answer
Factor the greatest common factor, or GCF, from each term a) The GCF is 5a. 5𝑎𝑚−5𝑎𝑛=5𝑎(𝑚−𝑛) b) The GCF is 1. 5 𝑥 3 −3 𝑦 2 is prime. c) The GCF is 2 𝑐 𝑐 4 −4 𝑐 3 +6 𝑐 2 =2 𝑐 2 ( 𝑐 2 −2𝑐+3) To check each factorization, multiply the two factors to find the original expression. Substitute numbers for the variables in the previous example and simplify. Compare the results. What do you find? The results are the same.

4. Example 2
Factor 𝑟 𝑡+1 +𝑥(𝑡+1).

5. Answer
Notice that 𝑟 𝑡+1 +𝑠 𝑡+1 contains the common binomial factor 𝑡+1. Think of 𝑡+1 as c. The expression can also be factored mentally if you apply the Distributive Property. 𝑟 𝑡+1 +𝑠 𝑡+1 =(𝑟+𝑠)(𝑡+1)

6. Answer You can visualize this factoring procedure with an area model.
𝑡+1
𝑡+1
𝑡+1 𝑟 𝑟 + 𝑠 = + 𝑠
𝑟(𝑡+1) +
𝑠(𝑡+1) =
(𝑟+𝑠)(𝑡+1)

7. Factor 𝑥 2 +𝑥+2𝑥+2 by grouping.
Example 3
Factor 𝑥 2 +𝑥+2𝑥+2 by grouping.

8. Answer
There are several way to factor this expression by grouping. One way is to group the terms that have a common coefficient or variable. Treat 𝑥 2 +𝑥 as one expression, and treat 2𝑥+2 as another expression. To check, multiply the factors.