3 Daily Standard & Essential Question MM1A2c :Add, subtract, multiply, and divide polynomialsMM1A2g: use area and volume models for polynomials arithmeticEssential Question: What are the three special products and how can you quickly find each one?
4 There are formulas (shortcuts) that work for certain polynomial multiplication problems. (a + b)2 = a2 + 2ab + b2 (a - b)2 = a2 – 2ab + b2 (a - b)(a + b) = a2 - b2Being able to use these formulas will help you in the future when you have to factor. If you do not remember the formulas, you can always multiply using distributive, FOIL, or the area model method.
5 Let’s try one! 1) Multiply: (x + 4)2 You can multiply this by rewriting this as (x + 4)(x + 4)ORYou can use the following rule as a shortcut:(a + b)2 = a2 + 2ab + b2For comparison, I’ll show you both ways.
6 Notice you have two of the same answer? 1) Multiply (x + 4)(x + 4)Notice you have two of the same answer?x2First terms:Outer terms:Inner terms:Last terms:Combine like terms.x2 +8x + 16x+4+4x+4xx2+4x+16+4x+16Now let’s do it with the shortcut!
7 1) Multiply: (x + 4)2 using (a + b)2 = a2 + 2ab + b2 That’s why the 2 is in the formula!1) Multiply: (x + 4)2 using (a + b)2 = a2 + 2ab + b2a is the first term, b is the second term(x + 4)2a = x and b = 4Plug into the formula a2 + 2ab + b2(x)2 + 2(x)(4) + (4)2Simplify.x2 + 8x+ 16This is the same answer!
8 2) Multiply: (3x + 2y)2 using (a + b)2 = a2 + 2ab + b2 a = 3x and b = 2yPlug into the formula a2 + 2ab + b2(3x)2 + 2(3x)(2y) + (2y)2Simplify9x2 + 12xy +4y2
9 Multiply: (x – 5)2 using (a – b)2 = a2 – 2ab + b2 Everything is the same except the signs! (x)2 – 2(x)(5) + (5)2x2 – 10x + 254) Multiply: (4x – y)2(4x)2 – 2(4x)(y) + (y)216x2 – 8xy + y2
10 Notice the middle terms eliminate each other! 5) Multiply (x – 3)(x + 3)Notice the middle terms eliminate each other!x2First terms:Outer terms:Inner terms:Last terms:Combine like terms.x2 – 9x-3+3+3x-3xx2-3x-9+3x-9This is called the difference of squares.
11 5) Multiply (x – 3)(x + 3) using (a – b)(a + b) = a2 – b2 You can only use this rule when the binomials are exactly the same except for the sign.(x – 3)(x + 3)a = x and b = 3(x)2 – (3)2x2 – 9
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