Bellwork 75÷15 21÷7 75÷25 72÷9 85÷5 132÷11 28÷7 144÷12 24÷6 180÷15

Bellwork 75÷15 = 5 21÷7 = 3 75÷25 = 3 72÷9 = 8 85÷5 = 17 132÷11 = 12 28÷7 = 4 144÷12 = 12 24÷6 = 4 180÷15 = 12

Algebra Factoring: GCF

Review 𝑥 2 𝑥 2 +2𝑥−6 𝑥 4 +2 𝑥 3 −6 𝑥 2 4𝑥 3 2𝑥 2 −3𝑥−5 𝑥 2 𝑥 2 +2𝑥−6 𝑥 4 +2 𝑥 3 −6 𝑥 2 4𝑥 3 2𝑥 2 −3𝑥−5 8 𝑥 5 −12 𝑥 4 −20 𝑥 3

Review 3𝑥 2 2𝑥 2 +2𝑥−4 6 𝑥 4 +6 𝑥 3 −12 𝑥 2 5𝑥 2 𝑥 2 −𝑥−7 3𝑥 2 2𝑥 2 +2𝑥−4 6 𝑥 4 +6 𝑥 3 −12 𝑥 2 5𝑥 2 𝑥 2 −𝑥−7 10 𝑥 3 −5 𝑥 2 −35𝑥

Notes: Factoring We know how this works: 6𝑥 3 𝑥 2 +3𝑥−5 18 𝑥 3 +18 𝑥 2 −30 But could we work backwards? The answer is yes! This is called factoring by Greatest Common Factor, or GCF.

Get Started Factoring: The first step to factoring by Greatest Common Factor is to identify the Greatest Common Factor. That is figure out the greatest “number” can divide all of the terms of the polynomial.

Factoring 3𝑥 2 +18𝑥−6 Identify the GCF: 3 Pull out this GCF and finish factoring the rest of the polynomial by “dividing” each term by the GCF: 3( 𝑥 2 +6𝑥−2)

Factoring 7𝑥 3 −28 𝑥 2 −56𝑥 Identify the GCF: 7𝑥 Finish the factoring: 7𝑥( 𝑥 2 −4𝑥−8)

Factoring 9𝑚 5 −81 𝑚 3 −90𝑚 Identify the GCF: 9𝑚 Finish it: 9𝑚( 𝑚 4 −9 𝑚 2 −10)

Factoring 35𝑚 7 −84 𝑚 3 −91 𝑚 2 Identify the GCF: 7 𝑚 2 Finish it: 7 𝑚 2 (5 𝑚 5 −12𝑚−13)

Working in Pairs 15𝑥 6 −35 𝑥 4 −55 𝑥 2 5 𝑥 2 (3 𝑥 4 −7 𝑥 2 −11) 11𝑥 5 −121 𝑥 2 −132𝑥 11𝑥( 𝑥 4 −11𝑥−12)

Notes What does the word prime mean? The general definition of a prime number, or in our case algebraic expression, is one which can only be divided (or factored) into itself and the number 1.

Example 2 𝑥 2 +14𝑥+3 This is prime because: 2, 14, and 3 have no common factor other than 1. There are no variables that can be pulled from all three terms

Example 5 𝑥 3 +2𝑥−9 PRIME

Working in Pairs: Prime Not Prime Are the following prime or not prime? 𝑥 3 +3𝑥−7 24𝑥 2 −32𝑥+4 28𝑥 2 +14𝑥−21 4𝑥 3 +5𝑥−9 Prime Not Prime

Brief Recap We can factor by Greatest Common Factor by: Identifying the GCF Then finishing the factoring by dividing the polynomial by the GCF A prime polynomial is one which can only be divided, or factored, by itself and the number 1.

Assignment 10-1

Notes: Factoring by Grouping Factoring by grouping works similar to factoring by GCF Instead of pulling out a GCF from all of the terms, we will split the polynomial into two parts and pull a GCF from each part

Example 4 𝑦 3 +8 𝑦 2 +5𝑦+10 Split it: 4 𝑦 3 +8 𝑦 2 5𝑦+10 Pull A GCF from each (if this is done right the parentheses should be the same: 4 𝑦 2 𝑦+2 +5 𝑦+2 Now re-write: (4 𝑦 2 +5)(𝑦+2)

Example 6 𝑦 3 −18 𝑦 2 +5𝑦−15 6 𝑦 3 −18 𝑦 2 5𝑦−15 6 𝑦 2 𝑦−3 +5 𝑦−3 6 𝑦 3 −18 𝑦 2 5𝑦−15 6 𝑦 2 𝑦−3 +5 𝑦−3 (6 𝑦 2 +5)(𝑦−3)

Example 𝑦 3 +7 𝑦 2 +3𝑦+21 𝑦 3 +7 𝑦 2 3𝑦+21 𝑦 2 𝑦+7 +3 𝑦+7 𝑦 3 +7 𝑦 2 3𝑦+21 𝑦 2 𝑦+7 +3 𝑦+7 ( 𝑦 2 +3)(𝑦+7)

Example 2 𝑚 3 +18 𝑚 2 −3𝑚−27 2 𝑚 3 +18 𝑚 2 −3𝑚−27 2 𝑚 2 𝑚+9 −3 𝑚+9 2 𝑚 3 +18 𝑚 2 −3𝑚−27 2 𝑚 2 𝑚+9 −3 𝑚+9 (2 𝑚 2 −3)(𝑚+9)

Assignment 10-2