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A1.7.4-5 Factor the following 36𝑥 2 − 121 4𝑥 4 − 64 Now we will learn to factor a quadratic expression A quadratic polynomial has the form: 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 Teach me how to factor There is a pattern that can help us factor : Multiply: (2𝑥 + 3)(𝑥 + 4) 2𝑥 + 3 𝑥 + 2𝑥 + 3 4 2𝑥 2 + 3𝑥 + 8𝑥 + 12 What is the product of the first and last term? What is the product of the middle two terms? Multiply (4x+6)(2x-4) 4𝑥 + 6 2𝑥 + (4𝑥 + 6)(−4) 8𝑥 2 + 12𝑥 − 16𝑥 − 24 What is the product of the first and last terms? What is the product of the middle terms? Before combining your like terms, the product of the middle terms will always equal the product of your first and last terms Who wants to see why? (ax+b)(cx+d) (𝑎𝑥 + 𝑏)𝑐𝑥 + (𝑎𝑥 + 𝑏)𝑑 𝑎𝑐𝑥 2 + 𝑏𝑐𝑥 + 𝑎𝑑𝑥 + 𝑏𝑑 What is the product of the first and last terms? 𝑎𝑐 ∗ 𝑏𝑑 = 𝑎𝑐𝑏𝑑 = 𝑎𝑏𝑐𝑑 What is the product of the middle terms? 𝑏𝑐 ∗ 𝑎𝑑 = 𝑏𝑐𝑎𝑑 = 𝑎𝑏𝑐𝑑 Multiplication is Commutative! 2 𝑎𝑥 + 𝑏𝑥 + 𝑐 Find 2 numbers that add to this That have the same product as these 2 When we find the right pair, we will split the middle term into two terms with those coefficients We are going to learn how to factor a quadratic to the product of 2 binomials (2𝑥 + 3)(𝑥 + 4) 2𝑥 + 3 𝑥 + 2𝑥 + 3 4 Today we will start with this 2𝑥 2 + 3𝑥 + 8𝑥 + 12 2𝑥 2 + 11𝑥 + 12 And turn it into this 2𝑥 2 + 7𝑥 + 5 What is the product of the first and last term? 2 ∙ 5 = 10 What are the factors of 10? 10 and 1, 5 and 2 Do either of these sets have a sum of 7? 5 and 2 Find the two middle terms 8𝑦 2 + 16𝑦 + 6 8 ∗ 6 = 48 Now that you found the two middle terms You can rewrite the quadratic with the new terms 8𝑦 2 + 12𝑦 + 4𝑦 + 6 48 factors 16 sum 1, 48 49 2, 24 3, 16 26 19 4, 12 16 If you are looking for two numbers that have a negative product what do you know about the sign of the two numbers? Ex: 𝑎 ∗ 𝑏 = −6 a is negative or b is negative (not both) If their sum is positive, the larger number is positive If their sum is negative, the larger number is negative What if the product is positive but the sum is negative? 𝑎 ∗ 𝑏 = 24 𝑎 + 𝑏 = −10 A and b are both negative! Find the middle two terms: 8𝑥 2 − 18𝑥 + 9 factors sum Find the middle two terms: 3𝑝2 − 25𝑝 + 16 factors sum Tomorrow we will learn how to factor after we split the middle term A.6.5 Day 1 Find the middle two terms for the following: 2𝑥 2 + 7𝑥 + 5 factors sum To factor this we will Factor by Grouping 2𝑥 2 + 5𝑥 + 2𝑥 + 5 Step 1: split the expression into two groups Step 2: Look at each group separately and factor out the GCF The GCF might be 1! 2𝑥 2 + 2𝑥 + 5𝑥 + 5 Both terms have an 2𝑥(𝑥 (𝑥 + 1). We can factor out an (𝑥 + 1) + 1) + 5(𝑥 + 1) (𝑥 + 1)(2𝑥 + 5) Factor each! 3𝑝2 + 8𝑝 + 5 15 factors 8 sum 7𝑝2 − 23𝑝 − 20 140 factors -23 sum 7𝑝2 − 24𝑝 + 9 63 factors -24 sum 6.5 day 2