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Objective The student will be able to: use patterns to multiply special binomials. SOL: A.2b Designed by Skip Tyler, Varina High School There are formulas (shortcuts) that work for certain polynomial multiplication problems. 2 b) 2 a 2 b (a + = + 2ab + (a - b)2 = a2 – 2ab + b2 2 2 (a - b)(a + b) = a - b Being 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 box method. Let’s try one! 1) Multiply: (x + 4)2 You can multiply this by rewriting this as (x + 4)(x + 4) OR You can use the following rule as a shortcut: (a + b)2 = a2 + 2ab + b2 For comparison, I’ll show you both ways. 1) Multiply (x + 4)(x + 4) First terms: x2 Outer terms: +4x Inner terms: +4x Last terms: +16 Combine like terms. x2 +8x + 16 Notice you have two of the same answer? x x +4 x2 +4x +4 +4x +16 Now let’s do it with the shortcut! 1) Multiply: (x + 4)2 That’s why the 2 is in the formula! using (a + b)2 = a2 + 2ab + b2 a is the first term, b is the second term (x + 4)2 a = x and b = 4 Plug into the formula a2 + 2ab + b2 (x)2 + 2(x)(4) + (4)2 This is the Simplify. same answer! x2 + 8x+ 16 2) Multiply: (3x + 2y)2 using (a + b)2 = a2 + 2ab + b2 (3x + 2y)2 a = 3x and b = 2y Plug into the formula a2 + 2ab + b2 (3x)2 + 2(3x)(2y) + (2y)2 Simplify 9x2 + 12xy +4y2 Multiply (2a + 3)2 1. 2. 3. 4. 4a2 – 9 4a2 + 9 4a2 + 36a + 9 4a2 + 12a + 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)2 x2 – 10x + 25 4) Multiply: (4x – y)2 (4x)2 – 2(4x)(y) + (y)2 16x2 – 8xy + y2 Multiply (x – y)2 1. 2. 3. 4. x2 + 2xy + y2 x2 – 2xy + y2 x2 + y2 x2 – y2 5) Multiply (x – 3)(x + 3) First terms: x2 Outer terms: +3x Inner terms: -3x Last terms: -9 Combine like terms. x2 – 9 Notice the middle terms eliminate each other! x -3 x2 -3x +3 +3x -9 x This is called the difference of squares. 5) Multiply (x – 3)(x + 3) using 2 2 (a – b)(a + b) = a – b 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)2 x2 – 9 6) Multiply: (y – 2)(y + 2) (y)2 – (2)2 y2 – 4 7) Multiply: (5a + 6b)(5a – 6b) (5a)2 – (6b)2 25a2 – 36b2 Multiply (4m – 3n)(4m + 3n) 1. 2. 3. 4. 16m2 – 9n2 16m2 + 9n2 16m2 – 24mn - 9n2 16m2 + 24mn + 9n2