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Special Products of Binomials You will be able to apply special products when multiplying binomials.

Being able to use these formulas will help you find products quickly. If you do not remember the formulas, you can always multiply using the distributive property, FOIL, or the box method.

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 = a 2 + 2ab + b 2 Lets examine both methods.

Multiply (x + 4)(x + 4) First terms: Outer terms: Inner terms: Last terms: Combine like terms. x 2 +8x + 16 x+4 x x 2 +4x +16 Let’s use the shortcut! x2x2 +4x +16 Notice you have two of the same answer?

Multiply: (x + 4) 2 using (a + b) 2 = a 2 + 2ab + b 2 a is the first term, b is the second term (x + 4) 2 a = x and b = 4 Plug into the formula a 2 + 2ab + b 2 (x) 2 + 2(x)(4) + (4) 2 Simplify. x 2 + 8x+ 16 This is the same answer! That’s why the 2 is in the formula!

Multiply: (3x + 2y) 2 using (a + b) 2 = a 2 + 2ab + b 2 (3x + 2y) 2 a = 3x and b = 2y Plug into the formula a 2 + 2ab + b 2 (3x) 2 + 2(3x)(2y) + (2y) 2 Simplify 9x xy +4y 2

Multiply (2a + 3) a 2 – a a a a a + 9

Multiply: (x – 5) 2 using (a – b) 2 = a 2 – 2ab + b 2 Everything is the same except the signs! (x) 2 – 2(x)(5) + (5) 2 x 2 – 10x + 25 Multiply: (4x – y) 2 (4x) 2 – 2(4x)(y) + (y) 2 16x 2 – 8xy + y 2

Multiply (x – y) 2 1. x 2 + 2xy + y 2 2. x 2 – 2xy + y 2 3. x 2 + y 2 4. x 2 – y 2

x2x2 +4x+4 x +2 x3x3 8 4x24x2 2x22x2 4x4x 8x8x This is called the cube of a binomial.

Multiply: (x + 2) 3 using (a + b) 3 = a 3 + 3a 2 b + 3ab 2 + b 3 a is the first term, b is the second term (x + 2) 3 a = x and b = 2 Plug into the formula a 3 + 3a 2 b + 3ab 2 + b 3 (x) 3 + 3(x) 2 (2) + 3(x)(2) 2 + (2) 3 Simplify. x x x + 8 This is the same answer!

Multiply (3a + 2) a a a a 2 + 6a a a a a a a + 8

Multiply: (x – 1) 3 using (a – b) 3 = a 3 – 3a 2 b + 3ab 2 – b 3 Everything is the same except the signs! (x) 3 – 3(x) 2 (1) + 3(x)(1) 2 – (1) 3 x 3 – 3x x – 1 Multiply: (x – 2y) 3 (x) 3 – 3(x) 2 (2y) + 3(x)(2y) 2 – (2y) 3 x 3 – 6x 2 y + 12 xy 2 – 8y 3

Multiply (x – 3)(x + 3) First terms: Outer terms: Inner terms: Last terms: Combine like terms. x 2 – 9 x-3 x +3 x 2 +3x -3x -9 This is called the difference of squares. x2x2 +3x -3x -9 Notice the middle terms eliminate each other!

Multiply (x – 3)(x + 3) using (a – b)(a + b) = a 2 – b 2 You can only use this rule when the binomials are conjugates. (x – 3)(x + 3) a = x and b = 3 (x) 2 – (3) 2 x 2 – 9

Multiply: (y – 2)(y + 2) (y) 2 – (2) 2 y 2 – 4 Multiply: (5a + 6b)(5a – 6b) (5a) 2 – (6b) 2 25a 2 – 36b 2

Multiply (4m – 3n)(4m + 3n) 1. 16m 2 – 9n m 2 + 9n m 2 – 24mn - 9n m mn + 9n 2